Thanks, have you thought about doing something similar with wages in other words if the Calvo thing was ever going to make any sense It feels like it would make a little bit more sense with wage setting You know you get your raise once a year, so I don't know if there was some scope for that I wonder if we show up there I would like to comment on the list of reasons why one wanted to have low inflation put up by Tony Ghani I Believe that something has been missing. So there's a six Reason and I think it is the most important one My background is German. So we have German history and the great inflation in the 20s destroyed the Weimar Republic and And so there is a feedback between the political system and Price stability In particular now that people live 20 years longer than they leave the labor force Which means they Somebody has to provide for their living after having left the labor force, which is by either Social Security provision or other government measures or by individual saving and If you rely completely on the government to provide for The living of the old generation the Government would be overwhelmed and this would probably would mean that the Society in a democratic way would no longer be able to operate. So we need The reliance on individual provision for one's own future That is we we need to rely on saving Individual saving and therefore it is particularly important from the political economy point of view that you provide a simple way of providing for the future and This is basically Price stability. I think you mentioned that cognitive Limitations of people in providing for the future may also be a reason to provide for Price stability and I believe this is a very important point This is a much more practical suggestion relating to Jordy's comment about the changes in prices are not equivalent to the cross-sectional variance So why might this go wrong because you might have a bunch of small changes period after period But if they're persistent then prices can slowly drift apart and so the changes might look small and the dispersion high and this might be Plausibly what happens in an inflation so my suggestion would be to correct with auto correlation That's something you probably could measure with your data because I get that like doing cross-section Directly is really hard because we've got a lot of different different products But the auto correlation within each product is something that you could compute and then changes over one minus auto correlation Is the cross-sectional variance of a stochastic process that has that variance in the innovation has that auto correlation? Does it solve all the problems? No, but would it get you know, bring us a lot closer to Narrowing that gap between changes and dispersion. Yeah, I think it would help So two reasons inflation was Was costly in the 70s was one Capital tax capital income taxation is nominal That's true now too. So if you increase Stabilized inflation you you increase the capital taxation Which you may or may not like but it has that implication second within with constrained financial markets with banks Essentially only should and with the loans high inflation meant that you have to repay that debt really fast with with That cost is probably not so large now given Financial innovation It's much easier to get around that particular cost of inflation, but I feel Capital taxation should not be forgotten And by the way, it would be nice if you if you emphasize Beyond besides steady-state inflation For central banks. What are they given that you you basically show that we should Trash implications of the Calvary model and and go with them and a new cost model what should Central banks think about in terms of the transmission mechanism there or the policies Well, thank you very much Jordy, I mean basically I agree with everything you say Let me talk a little bit about this relationship between absolute size and measures of inefficient price dispersion It's it's absolutely right that this is an indirect link, you know, this is you know something Laura also mentioned as well The reason that this is so challenging is that I think a very convincing conclusion of the empirical literature on pricing in recent years Is that you have to consider idiosyncratic shocks the model? Just doesn't work at all in terms of fitting the data if you don't allow for some kind of idiosyncratic shocks to firms desired prices because Average prices are 10% average price changes or something like 10% per year And so there must be a lot of unobserved reasons why firms want to change their prices that aren't related to average inflation And then once you allow for the fact that there's a lot of unobserved reasons for why firms want to change their prices Then it becomes much more difficult to tell whether they have the right price or the wrong price Like for example, if you didn't have these idiosyncratic shocks You could just look at how far the current price was from some kind of long-term average Like a fixed effect regression or something and that would give you some sense of inefficient price dispersion But if you have these idiosyncratic shocks then when the firm changes its price It might be because they really wanted to because something happened that made it want to change its price And so our approach of using these absolute size of price changes kind of like a revealed choice approach The idea is that if you if you did decide to change your price By a lot then that indicates that your initial price before you adjusted was was far off So that's kind of the idea that we're using this kind of revealed preference approach But it's absolutely true that this relies on some structure So in particular it relies on the notion that is true in many of these models, but not universally true That you know your your your difference between your current price and your desired price would be related to how much you adjust Conditional and adjustment So for example in a world where you always change your price by five percent no matter what no matter what then our method Really wouldn't work because even if you're super far from your desired price You still change it by just five percent per year, you know or whatever per period every time and it just doesn't work So that that I think is is is an important assumption in what we're doing More generally I think this role of idiosyncratic shocks is a huge difference between this class of models and the earlier generation of models So, you know the earlier generation of models which didn't have these idiosyncratic shocks There is an earlier generation of menu cost models where in fact the SS bands themselves were pretty sensitive to steady-state inflation And and that was basically because there weren't these idiosyncratic shocks But I you know I think it is a genuine challenge of how to come up with a with a measure And so this is sort of a semi structural approach to doing it I totally agree with the idea that there there are other things like for example cognitive costs and Political concerns and so on that we need to think about it's important to recognize that there are other potential upsides of higher inflation like for example if there are sticky wages There might be there's this idea that having higher inflation might sort of grease the wheels of the labor market And that could be a potential upside on this issue of whether the Caval model already sort of generates essentially nothing at low inflation rates It's worth recognizing that the axes on that graph make the changes at low inflation rates look really small But they are something like even at low inflation rates are like half a percent of Steady-state, you know consumption so depending on how you specify things that could play a role And I I mean I'm very sympathetic to what you said about your model suggesting that these costs were small relative to the potential cost of hitting the zero lower bound But there is other research where the cost of hitting the zero lower bound maybe is Parameterized to be smaller and so even this half a percent of steady-state consumption actually makes a difference Thank you. Thanks a lot


My name is Sam Vatmy. I'm the author of Malignan self-love, Narcissism-Privileged. Today I would like to discuss a model of empathy. In my work, I suggest that empathy is three-parted, as three parts, roughly corresponding to Freud's postulated ego and superego. In this model, normal empathy is comprised of three components, instinctual, emotional, and cognitive. Children develop empathy in three phases, which correspond to these three components. Children construct the emotional and cognitive tears upon an instinctual firmament. In adults, cognitive empathy always goes hand-in-hand with the instinctual element and with the emotional correlate or component. But what is this empathy that I'm talking about? What is empathy? Normal people use a variety of abstract concepts and psychological constructs to relate to other people. Emotions are such modes of interrelatedness. Narcissists and psychopaths are different. Their equipment is lacking. They understand only one language, self-interest. Their inner dialogue and private language revolve around the constant measurement of utility. They regard others as mere objects, instruments of gratification, and representations of functions. This deficiency in the narcissist and psychopath renders them rigid and socially dysfunctional. They don't bold. They become dependent on narcissistic supply, on drugs, on adrenaline rushes, on money, on power. They seek pleasure by manipulating their theorist and nearest, or even by destroying them, the way a child interacts with historians.  Their interlocutor's body language, the subtleties of speech, social etiquette, they're all lost of them. Narcissists and psychopaths lack empathy. Empathy requires both the suspension of disbelief by assuming someone else's identity, same way actors do, and the surrender of control by allowing other people to dictate how one feels. Both themes, both feats, go against the grain of narcissists, let alone psychopaths. They are unable to suspend disbelief, and they definitely are not about to hand-control to anyone or anything. It is safe to say that the same applies to such patients who are co-diaglos, co-mormin, with other personality disorders, notably schizotyp, paranoid, borderline avoidant, and schizotypo. Empathy lubricates the wheels of interpersonal relationships. Encyclopedia Britannica defines empathy as the ability to imagine oneself in another's place and understand the other's feelings, desires, ideas, and actions. It is a term coined in the early 20th century, equivalent to the German Einführung, and modeled on sympathy. The term is used with special but not exclusive reference to aesthetic experience. The most obvious example, perhaps, is that of the actor or singer who genuinely feels the part he is performing. With other works of art, the spectator may, by a kind of introjection, feel himself involved in what he observes, or contemplates. The use of empathy is an important part of the counselling technique developed by the American psychologist Carl Rogers, end of quote. And this is how empathy is defined in Psychology and Introduction by Charles G. Morris. He says, empathy is closely related to the ability to read other people's emotions. We arousal of an emotion in an observer at least vicarious, a vicarious response to other people's situation. Empathy depends not only on one's ability to identify someone else's emotions, but also on one's capacity to put oneself in the other person's place, and to experience an appropriate emotional response. Just as sensitivity to non-verbal cues increases with age, so does empathy. The cognitive and perceptual abilities required for empathy developed only as a child matures. In empathy training, for example, each member of the couple is taught to share inner feelings and to listen to and understand the partner's feelings before responding to them. The empathy technique focuses the couple's attention on feelings and requires that they spend more time listening and less time in rebuttal. Empathy is also a cornerstone of morality, again the Encyclopedia Britannica. Empathy and other forms of social awareness are important in the development of amoral sets. Morality embraces a person's beliefs about the appropriateness or goodness of what he does, thinks or feels. Childhood is the time at which moral standards begin to develop in a process that often extends well into adulthood. The American psychologist Lorenz Korberg hypothesized that people's development of moral standards passes through stages that can be grouped into three moral levels. At the third level, that of post-conventional moral reasoning, the adult bases his moral standards on principles that he himself has evaluated and that he accepts as inherently valid, regardless of society's opinion. He is aware of the arbitrary subjective nature of social standards and rules which he regards as relative rather than absolute in authority. Thus, says the Britannica, the basis for justifying moral standards pass from avoidance of punishment to avoidance of adult disapproval and rejection to avoidance of internal guilt and self-recrimination. The person's moral reasoning also moves towards increasingly greater social scope, in other words including more people in institutions and greater abstraction, in other words from reasoning about physical events such as pain or pleasure to reasoning about values, rights and implicit conflicts. The Britannica proceeds, others are valued that because even rather young children are capable of showing empathy with the pain of others, the inhibition of aggressive behavior arises from this moral effect rather than from mere dissipation of punishment. Some societies around the children differ in their individual capacity for empathy and therefore some children are more sensitive to moral prohibitions than others. Young children's growing awareness of their own emotional states, characteristics and abilities leads to empathy, in other words to the ability to appreciate the feelings and perspectives of others. Empathy and other forms of social awareness are in turn important in the development of a moral sense. Another important aspect of children's emotional development is the formation of their self-concept or identity, their sense of who they are and what their relation to other people is. According to Lipp's concept of empathy, a person appreciates another person's reaction by a projection of the self onto the other. In his aesthetic book, Lipp made all appreciation of art dependent upon a similar self-projection into the ocean. And so these are the views about empathy. But is empathy a social conditioning, a result of social conditioning or is it in bread, is it an instinct, a reflexive? Well, this question may well be the key to all art deliberations. Empathy has little to do with the person with whom we empathize, the empathy. It may simply be the result of conditioning and socialization. In other words, when we hurt someone, we don't actually experience his or her pain. We experience our pain. Hurting somebody hurts us. The reaction of pain is provoked in us by our own actions. We have been taught a learned response to free pain when we inflict pain. We attribute feeling sensations and experiences to the object of our actions, which is a psychological defense mechanism of projection. Unable to conceive of inflicting pain upon ourselves, we displace the source of the pain. It is the other's pain that we are feeling, we keep telling ourselves. It is not our pain, we did not do it to ourselves. Additionally, we have been taught to feel responsible for our fellow beings, and this is what we call guilt. So we also experience pain whenever another person claims to be ambushed. We feel guilty going to his or her condition. We feel somehow responsible or accountable, even if we had nothing to do with the whole affair. In some, we use the example of pain when we see someone hurting because we experience pain for two reasons. We feel guilty or somehow responsible for his or her condition, and it is a learned response. We experience our own pain and project it onto the empathy the person we empathize with. We communicate our reaction to the other person and agree that we both are sharing the same feeling of being hurt or being in pain, for example. This unwritten and unspoken agreement is what we call empathy. Back to the Encyclopedia Britanta. Perhaps the most important aspect of children's emotional development is a growing awareness of their own emotional states and the ability to discern and interpret the emotions of others. The last half of the second year is a time when children start becoming aware of their own emotional states, characteristics, abilities and potential for action. This phenomenon is called selfish awareness, coupled with strong narcissistic behaviors and traits. The Britannica says, this growing awareness and ability to recall one's own emotional states leads to empathy or the ability to appreciate the feelings and perceptions of others. Young children's dawning awareness of their own potential for action inspires them to try to direct or otherwise afflict the behaviors of others. With age, children acquire the ability to understand the perspective or point of view of other people, a development that is closely linked with the empathic sharing of other people's emotions. One major factor underlying these changes is the child's increasing cognitive sophistication. For example, in order to fill the emotion of guilt, a child must appreciate the fact that he could have inhibited a particular action of his that violated a moral standard. The awareness that one can impose a restraint on one's own behavior requires a certain level of cognitive maturation, and therefore the emotion of guilt cannot appear until the competence, that competence, that cognitive competence is attained. Still, empathy maybe an instinctual reaction to external stimuli that is fully contained within the empath or the person who empathizes, and then projected onto the empathy the person he empathizes with. This is clearly demonstrated by inborn empathy. It is the ability to exhibit empathy and altruistic behavior in response to facial expressions. Even newborns react this way to their mother's facial expression of sadness or distress. And this serves to prove that empathy has very little to do with feelings, experiences or sensations of the other, the empathy. Surely the infant has no idea what it is like to feel sad, and definitely not what it is like for his mother to feel sad. In this case it is a complex, reflexive reaction, a reflex. Later on empathy is still rather reflexive, the result of conditioning. The Britannica quoted some fascinating research that supports the model that I propose. An extensive series of studies indicated that positive emotion feelings enhance empathy and altruism. It was shown by the American psychologist Alice M. Eisen, the relatively small favorites of a piece of good luck like finding money in a coin telephone or getting an unexpected hit, induced positive emotion in people, and that such emotion regularly increased the subject's inclination to sympathize or to provide help. Several studies have demonstrated that positive emotion facilitates creative problem solving. One of these studies showed that positive emotion enabled subjects to name more users for common objects. Another showed that positive emotion enhanced creative problem solving by enabling subjects to see relations among objects, and therefore among people, I say. Such relations would otherwise go unnoticed. A number of studies have demonstrated the beneficial effects of positive emotion and thinking, memory, and action in preschool and older children. If empathy increases with positive emotion, then it has little to do with the empathy the person we empathize with, the recipient or object of empathy, and everything to do with the empathore, the person who does the empathize. As Paul Bloom notes in his Contrarian Essay, The Baby in the Well, published in The New Yorker, empathy is a blunt, biased, and stereotypical tool is suited for guiding the design of public policy which ought to be partial to justice and constructive outcomes. We bestow our empathy on those who most resemble us, and on identifiable victims who garner the most media attention. Empathy for individual sufferers blinds us to the overall pictures, and provokes in us the base instincts of retribution and vengeance. It distorts decision making, thinking with one's heart, rather than with one's mind, is bound to yield catastrophic consequences. And this is precisely why we delegate the weighing of empathy and its implementation to faceless bureaucratic institutions. They are less likely to be swayed by prejudice and preconception. They are more likely to optimize resources. In the long run, such institutions benefit the many, not the few. But back to the very beginning. I said that narcissists and psychopaths lack empathy, but that's not precisely true. Contrary to widely held views, narcissists and psychopaths may actually possess empathy. They may even be hyper-empathic, attuned to the minutest signals emitted by their victims. They may be endowed with what I call a penetrating X-ray vision. Psychopaths and narcissists tend to abuse their empathic skills by employing them exclusively for personal gain, for the extraction of narcissistic supply or in the pursuit of anti-social and sadistic goals. They regard their ability to empathize as another weapon in their arsenal. There are two possible pathological reactions to childhood abuse and trauma. Codependence and narcissism.  The codependent has a pretty realistic assessment of herself, but her view of others is fantastic. The narcissists' self-image and self-perception are delusional and grandiose, but his penetrating view of others is blood-curdingly accurate. And so therefore I suggest to label the narcissistic psychopath's version of empathy called empathy, akin to the cold emotions felt by psychopaths. The cognitive element of empathy is there, but not its emotional correlate. It is consequently a barren, detached and cerebral kind of intrusive gaze devoid of compassion and of a feeling of affinity with one's fellow humans. Cold empathy is not the same as merely cognitive empathy. It is intuitive. It is the residual instinctual component coupled with cognitive empathy, but divorced from and leapfrogging the emotional constituent. So cold empathy equals instinctual empathy plus cognitive empathy minus emotional empathy. Cold empathy is the ossified consequence of arrested empathy. It is a predator of sympathy. It is all about resonance, not about putting yourself in other people's shoes. Narcissists and psychopaths also appear to be empathizing with their possessions, for instance. Objects, pets, the sources of narcissistic supply or material benefits. It may be their nearest and dearest, significant others, friends or associates, but it may be ossified. This is not real empathy. It is a mere projection of the narcissists of psychopath's own insecurities and fears, needs and wishes, fantasies and priorities. This kind of display, sometimes postentatious on empathy, usually vanishes. The minute the subject sees us to play a role in the narcissists or psychopath's life and his psychodynamic processes, the minute the source of supply, for instance, sees us to provide the narcissists with narcissistic supply, all vestiges of empathy vanish. Cold empathy evokes the concept of uncanny valley, coined in 1970 by the Japanese roboticist Masahiro Morin. Morin suggested that people react positively to androids, human-like robots. For as long as these androids differ from real humans in meaningful and discernible ways, but the minute these contractions, these gadgets, these devices come to resemble humans uncannyly, though imperfectly, humans, human observers tend to experience repulsion, revulsion and other negative emotions, including fear. And the same applies to psychopathic narcissists. They are near perfect limitations of human beings, but lacking empathy and emotions, they are not exactly there. Psychopaths and narcissists strive their interlocutors as being some kind of alien life forms or artificial intelligence, in short, akin to humanoid robots, to androids. When people come across narcissists and psychopaths, the uncanny valley reaction kicks in. People feel ill it is, revolted, scared, repelled. They can't put their finger on what it is that bothers them, what it is that provokes these negative reactions. But after a few initial encounters, they tend to shy away and to keep their distance. But how important is empathy to proper psychological functioning? Why can't the narcissists and psychopaths be normal, despite the fact that they don't have empathy? Well, empathy is more important socially than it is psychologically. The absence of empathy, for instance, as I said in narcissistic or antisocial personality disorder, the absence of empathy predisposes people to exploit and abuse others. Empathy is the bedrock of our sense of morality. Arguably aggressive behavior is as inhibited by empathy at least as much as it is by anticipated punishment. But the existence of empathy in a person is also a sign of self-awareness. A healthy identity, a well-regulated sense of self-worth and self-love in the positive sense. Its absence denotes emotional and cognitive deficits or immaturity. An inability to love, to truly relate to others, to respect other people's boundaries, to accept their needs, feelings, hopes, fears, choices and preferences as autonomous entities. How is empathy developed? Empathy may be innate. Even toddlers seem to empathize with the pain or happiness of others, such as their caregivers. Empathy increases as the child forms a self-concept and identity. The more aware the infant is of his or her emotional states, the more he explores his limitations and capabilities, the more crony is to projecting this newfound knowledge onto others. But attributing to people around him his new gained insights about himself, by doing so, the child develops a moral sense and inhibits his anti-social impulses. The development of empathy is therefore a part of the process of socialization and self-identity. But as the American psychologist Carl Rogers taught us, empathy is also learned and inculcated. We are coached to feel guilt and pain when we inflict suffering on another person. Empathy is an attempt to avoid our own self-imposed agony by projecting it onto another. Is there an increasing dearth of empathy in society today? Well, the social institutions that reified, propagated and administered empathy have imploded the nuclear family, the closely net extended clan, the village, the neighborhood, the church have all unraveled. Society is atomized and anomic. The resulting alienation fostered a wave of anti-social behavior, both criminal and legitimate, quote-unquote. The survival value of empathy is on the decline. It is far wiser to be cunning to cut corners, to deceive and to abuse than to be empathic. Empathy has largely dropped from the contemporary curriculum of socialization. In a desperate attempt to cope with these inexorable processes, behaviors predicated on the lack of empathy have been pathologized and medicalized. The sad truth is that narcissistic or anti-social conduct is both normative and rational in contemporary society. No amount of diagnosis, treatment and medication can hide or reverse this fact. Ours is a cultural malaise which permeates every single cell and strand of the social fabric. But is there any empirical evidence that we can point to of a decline in empathy? Empathy cannot be measured directly only through proxies such as criminality, terrorism, charity, violence, anti-social behavior, related mental health disorders, abuse. It is also extremely difficult to separate the effects of deterrence from the effects of empathy. If I don't batter my wife, if I don't torture animals, I don't steal, is it because I'm empathetic or because I don't want to go to jail? No, no. Rising litigiousness, zero tolerance and skyrocketing grace of incarceration, as well as the aging of a population, have sliced intimate partner violence and other forms of crime across the United States at least in the last decade. But this benevolent decline has nothing to do with increasing empathy as to with demographic creatures. The statistics are open to interpretation, but it would be safe to say that the last century has been the most violent and least empathetic in human history. Wars and terrorism are on the rise, charity giving on the wane, measured as percentage of national wealth, welfare policies are being abolished, Darwinian models of capitalism are spreading, income inequality is gaping. In the last two decades, mental health disorders were added to the Diagnostic and Statistical Manual of the American Psychiatric Association, whose hallmark is the lack of empathy. The violence is reflected in our popular culture, movies, video games and media. Empathy, supposedly a spontaneous reaction to the plight of our fellow humans, is now channeled through self-interested and bloated non-government organizations or multilateral outfits with great tax credits. The vibrant world of private empathy has been replaced by faceless state largesc, pity, mercy, the elation of giving are all tax deductible, and it is a sorry sight indeed. We live in the narcissist world.


Welcome back to our lecture series, MAT3120 Transition to Advanced Mathematics for Student Sets at the Utah University. As usual, I'll be your professor today, Dr. Anjan Misildine. Lecture 10 represents the start of the second unit of our lecture series. Lectures 1 through 9 focused on this idea of set theory, for which we're introduced to the notions of sets, subsets, partitions, elementary power sets, and just elementary operations on sets, unions, intersections, Cartesian projects, and such. Of course, with this lecture series, this runs concurrently with our conversations about logic and communication. But for the mathematical content, lectures 1 through 9 cover this idea of elementary set theory. Now starting with lecture 10, we're going to be starting the conversation about a topic of mathematics known as commonatorics. With commonatorics, there's a lot of ways you could define it. Basically, commonatorics, for at least the purposes of our lecture series, its scope is much larger than what we're going to talk about. But for this lecture series, commonatorics is the mathematics of counting things. That might not seem like it's a hard thing. It's like, I learned to count before I even went to school. One, two, three. Well, it's a little bit more complicated than that. Commonatorics focuses on the idea of a finite set. You have some set A, and you then look at the cardinality of that set. When it's finite, we want a specific number. Is it 7? Is it 3,000? Is it 5,280? It turns out that there are a lot of tricky things that can happen when you're trying to compute the cardinality of a finite set. Commonatorics is focused on that. We will continue to have a conversation in these lecture series about logic and communication that are concurrent with our mathematical conversation. But for the topics at hand, the mathematical content is going to shift towards this idea of commonatorics. As we talk about commonatorics, we're going to be very interested in counting sets and lists and things like that. In this first video, we want to focus on perhaps the most fundamental principle of commonatorics, which is sometimes referred to as the multiplication principle. This is because in the future, we'll also talk about the addition, subtraction, and division principles of commonatorics. So before we see what the multiplication principle is, let's actually look at an example. So let's imagine that you are going to go eat dinner at the following local restaurant. And this local restaurant's really attractive to say college students because every meal on the menu is the exact same price, right? There's the student fixed price meal. And this is how it breaks down. You get an appetizer with this meal. It comes with a single entree and it comes with a single dessert.  But now that price is no longer the issue, it comes down to what do you pick for your meal, right? We could ask with these three choices that have to be made. You have to choose an appetizer. You have to choose an entree. You have to choose a dessert. How many different meals could be ordered using the fixed price dinner at this restaurant? Well, this is a small enough problem that we could really consider all the possibilities. You have two choices for this appetizer. You have three choices for the entree, right? Cause appetizers, you had soup and salad, those were your two options. For entree, you could have lasagna, steak, curry or lobster. Four options there. And then for dessert, you get to choose between ice cream or cheesecake. Now, to help us prepare for future combinatorial problems, I want you to think of it the following way. We're gonna introduce a set, which we're gonna call A, A for appetizer. And this set contains two elements. There's soup and then there's salad. There is a second set, which we're gonna call it E, E for entree there.  And then for dessert, we have two options, of course. There's the ice cream and then there's the cheesecake. I wish I could pick both, but alas, you only get one option of each of these things here. So we think of these as sets, right? So for each set, each set represents a decision that has to be made, a choice that has to be made. I'm gonna put it more as decisions here, because the word choice can be a little bit ambiguous in this situation. So we have a set for these three decisions that have to be made. Now, inside of each of these decision sets, there are options. The elements of these sets are gonna be the options you can have there. And therefore, if you look at something like the cardinality of set A, that's gonna be the number of options for appetizers. And likewise, if you look at the cardinality of E here, that's gonna give you the number of options for the second decision of entree. And if you look at the cardinality of D, that will be the number of options you have for the dessert. So these cardinalities are gonna affect the number of meals you can have here. Now, let's think about what is a meal? In this abstract sense, a meal is actually a list of three things. You pick your appetizer, let's say it's a soup. You pick your entree, let's say it's the lasagna, right, lasagna. And then let's say you pick your dessert and you pick the ice cream for that one, right? And so a meal is a list of three things. It's an object from the first set, decision set. There's an element chosen from the second decision set. And then there's an element chosen for the third decision set. So a meal combination, we can then view as an element of the Cartesian product, appetizers times entrees times desserts there. And so, because a meal is just a list of three elements, one appetizer, one entree and one dessert. And therefore the number of meal combinations, the number of meal combinations is then equal to the cardinality of this set, A cross E cross D. And we've talked about this before for two Cartesian sets. And so if you look at this like, oh, this set is really just A cross E cross D, like so. And so therefore the cardinality with two sets, you're gonna get A times the cardinality of E cross D, like so. But then once the cardinality of E cross D there, it's gonna be A times E times, got a little bit stuck there, times D. You take the product of all those three things there. So the number of meals is then gonna turn out to be two times four times, did I never write this on the board earlier too? You get two times four times two. So the number of meals is gonna be 16. And where did that two, four, two come from? That's the cardinality of this set, times the cardinality of this set, times the cardinality of this set. Now this seems like an overkill for what could be otherwise considered a very elementary problem. But the idea is we want to understand the principles and play here so that when we look at much more difficult examples, we can apply those principles in an abstract setting. So this fixed price local restaurant meal combination is an example of the multiplicative principle of counting that I alluded to earlier. So imagine we have a sequence of decision sets. So D one, D two, D three up to D N. So these sets represent decisions that have to be made and D I therefore is then the Ith decision that has to be made. Now let's suppose that each of these decisions is independent, which means that the choice of one decision doesn't actually affect the choice of another decision. So with our example of the restaurant, if you choose a soup that doesn't change your four options, lasagna, steak, curry or lobster. Now you the customer will be like, ah, soup goes better with a lobster. Curry is basically soup already, I don't wanna do that. Well, you might choose to not have curry because it's too similar to soup. But in all reality, the restaurant will let you do that combination. You could do soup and curry, making one choice doesn't stop the other. So these choices are independent, the decisions that is. And so then if you have N decisions that have to be made and those decisions are independent, then the number of possible choices you can make is gonna be the cardinality of the Cartesian product of all of those decisions together. Because like with the meals we talked about earlier, choosing a meal combination is selecting a list from that Cartesian product. So therefore we have to count the Cartesian product here. But how do you count the Cartesian product when each of these decision sets is finite? Well, it's going to be the product of each of these decision sets individually, the cardinalities that is. And so that's the multiplicative principle, but we're gonna use this all the time in common at Torx. Let me give you another example that's a lot quicker than the previous one. How many two symbol code words can be formed if the first symbol is an uppercase letter, so A through Z, Roman alphabet there, 26 letters. And the second symbol is gonna be a numerical digit, zero through nine. So there's 10 options for that one. So some examples like A1, A2, B3, X0, those are all examples of these two symbol code words. We could try to list each and every one of these code words, but that's not the best idea. Instead we wanna think of that each of these code words is a list of two elements. And we have basically two alphabets in play here. The first decision is we have to choose a letter. So we have some alphabet, which we'll call this A here. It contains the letters A, B, C, D, E, F, G, you know the song, right? So you have this alphabet for the first symbol. And then you have this alphabet for the second symbol. In this case, it's a different alphabet. The second symbol is then a numerical digit, zero, one, two, all the way up to nine. And like I mentioned earlier, the first alphabet, whoops, has a cardinality of 26. The second alphabet has a cardinality of 10. And the code words we're selecting is an element of the first alphabet cross the second alphabet. I keep on calling it alphabets because we're selecting a symbol, but an alphabet in this case is just our decision set. So the number of possibilities is the number of code words is then gonna be the cardinality of the Cartesian product, which we saw from the multiplicative principle. This is 26 times 10. So therefore there's gonna be 260 possible code words. That's all there has to be, 26 times 10 there. Let's look at another example. Consider the list of length four made with the symbols A, B, C, D, E, F, G, okay? Now this is something that's very important to pay attention to this. Now when we talk about these list of order four, remember, list are ordered. It's not a set at the list. The order matters. Now with the two previous examples we saw, the alphabets, the decision sets were different for each decision. So with appetizers, entrees, desserts, each one was different. So if you mix up the order, it didn't make any bit of a difference whatsoever. Same thing with our code words we had a moment ago. It was a Roman letter, then a number digit. So if you mix up the order, you could correct it without any problem. But the issue you have is like if your word is something like rat, okay? If you switch up the letters, you can also spell art, but like in the English language, rats and art are very different. Like if you ask me, let's go to the museum and look at some art, I might say, yeah, that sounds fun. But if you're like, let's go to the museum and look at some rats, I'd probably be like, no, I'm not gonna do that. Doesn't sound very fun. List the order matters. And so with this list of order four, excuse me, of length four, the order in which the letters appear will matter. If you permute things around, that gives you a different list. So we have to consider that when you repeat the alphabet. Another thing that's gonna be very important as you count list is this idea of repetition. This is an ordered list, the order matters, because that's built into the definition of the word list there. But this idea of repetition also matters. Can you use the same letter more than once? Well, again, when your different alphabets are, well, when the distinct alphabets are actually different from each other, then you have no chance of repetition because their intersections are empty. But in this situation, we're considering with repetition to find these lists. We have to take our alphabet. Let's take this right here to be our alphabet. Call it A again. And so a list of length four, for which you can repeat the same letter, this is looking for an element, such one of these four lists is gonna be an element of A, whoops, A cross A, cross A, cross A. Which remember, we defined this earlier, this is the set A to the fourth, right there. The exponent means how many Cartesian products did you take there? And so we wanna find the cardinality of this thing. The cardinality of A to the fourth by the multiplicative principle is gonna be the cardinality of A to the fourth power. All right, and how many letters are in our alphabet? One, two, three, four, five, six, seven. So we end up with seven to the fourth as the number of length four lists we can make with seven letters here. And of course, seven to the fourth, it turns out to be 2,401. Honestly, when it comes to combinatorial problems, the unsimplified answer is typically more enlightening than the simplified answer. So don't be too worried if you leave it as something like seven to the fourth there. Anyone can operate a calculator and figure out that number. So it's not a big deal if you leave it as something like seven to the fourth. And so the multiplicative principle comes into play here because as you're listing these things, you have basically four buckets that have to be filled. The first bucket has seven options. Because of repetition, this next bucket has seven options. You could choose the same letter as many times as you want. And so you have this seven times seven times seven and that's where the seven to the fourth comes from. Now I wanna change this problem ever so slightly because we still wanna consider list of length four but what if we don't allow for repetition anymore? So we can't reuse the same letter that we used previously. Now the reason why this example does need to be considered is because if repetition is not allowed, then it would appear that the decisions we make about which letters are in the list no longer are independent. Because if your first letter is A, then the second letter can't be A. So it kind of depends on the previous choice. Since we no longer have independent decisions, it would appear that the multiplication principle doesn't apply here but actually it does. And so what we're gonna do is the following. We're gonna introduce some new sets. So we're gonna introduce a set A1 which is just gonna be the alphabet we have from before, the same seven letters, A through G it was. Seven letters there. Now for the first symbol, you have seven options for what you can choose. Seven options there. Because you haven't used any of the seven letters yet. So that gives you your first letter. Now let's say for the sake of example, you choose an A. Well then what do you choose for the next letter? Well you choose everything except for A but I don't actually know what that letter is. So we're gonna say the first letter is, we'll call it X. I guess I'm gonna use more of the more than one. We'll call it X1. So the first letter whatever it is is X1. It could be an A, it could be a B, it could be C. Don't know. Now when it comes to the second letter, now when it comes to the second letter that you have to choose, you no longer have all the same options but what you can do is you take A1 and if you take away from it X1, so you remove X1 from the alphabet, you could then choose any letter in that set. We're gonna call this second alphabet A2. Now A2, because you took away a letter, its cardinality is actually now six and so it got smaller. You have one fewer options for the next one. Let's say that the second letter you choose is X2. Then we can repeat this pattern here that your third letter will be chosen from the set A2 take away X2. That's our third letter that we get to choose. Let's say that the third letter we choose is in fact X3. Well, how many options were there for X3? Well, it's the cardinality of A3 but now there's five elements there and we can then repeat this pattern for the fourth letter. We'll call that set A4. It's gonna have a cardinality of four and let's suppose that fourth letter we chose was X4 here. It's just a variable there but what was A4? A4 was of course A3 take away X3. Now be aware that X2, A2 was already missing X1 so when you take away X2 it's a different letter. You can't repeat the letter. So each time this thing gets smaller by one and we're never repeating the same letter there. So the type of words that we're looking for is you're looking for some type of list that's contained inside of A1 cross A2 cross A3 cross A4. Whoops, that is not an A. Try that again. And therefore what we have to do is we have to count the cardinality of A1 cross A2 cross A3 cross A4 for which by the multiplication principle because we've removed elements from the set once you remove the letter you've already used then the next decision is in fact independent of the previous ones. And so we end up getting this product seven times six times five times four and that turns out to be 840 words without repetition in there. And so with the appropriate modification we can in fact use the multiplication principle. And I drew these pictures of buckets before as we're trying to pick a word it's basically like we still have these four buckets that have to be filled and I'm choosing letters from the alphabet. Now this time we had seven available letters for the first one but depending upon what your first choice is then you have six options for the second and depending on that choice you get five options for the third and then the last one will be four options because you keep on using them. So even if you have some dependent choices if you have no repetition you actually get this falling product and instead of getting seven to the fourth now you actually are falling down what we refer to as a falling factorial but this is something we'll talk about more in a future lecture.


Ukrainian border guards destroyed an entire field camp of Russian occupiers where they stored anti-tank mines according to the State Border Guard Service of Ukraine. A magical fiery spectacle, the enemy's field depot of anti-tank mines soared into the air, declared the border guards in the post accompanying the video. As reported by the State Border Guard Service, aerial reconnaissance border guards in the south discovered a camouflaged field depot of anti-tank mines. Alongside fellow artillery men, the defenders destroyed it, stated the State Border Guard Service. In the Svatov sector, Ukrainian border guards destroyed five enemy equipment units including an armoured personnel carrier and an armoured fighting vehicle as well as two ammunition depots using combat drones. In the Svatov direction, the border guards, with the help of combat drones, destroyed five units of enemy equipment including an armoured personnel carrier and an armoured fighting vehicle. Two ammunition depots were blown up. A self-propelled artillery unit, an amphibious armoured scout car, an armoured fighting vehicle, three cars, and URAL truck were damaged and the invaders' shelters were also damaged by fire, the report said. In total for the week from November the 20th to the 26th, Ukrainian military forces eliminated 6,260 Russian soldiers. The armed forces of Ukraine also successfully destroyed 146 unmanned aerial vehicles, 130 artillery systems and 78 tanks.


Russia has amassed a large missile stockpile ahead of winter, and we see new attempts to strike Ukraine's power grid and energy infrastructure, trying to leave Ukraine in the dark and cold. Russia is also becoming much more dependent on China. Year by year, Moscow is mortgaging its future to Beijing. Militarily, Russia has lost a substantial part of its conventional forces. Hundreds of aircraft, thousands of tanks, and more than 300,000 casualties. We are now to outline the timeframe for the actual supply of fighter jets to Ukraine. Allies agree that Ukraine will become a member of NATO. We have now provided recommendations on Ukraine's priority reforms, including the fight against corruption, strengthening the rule of law, and supporting human rights and minority rights. Ukraine is closer to NATO than ever before. We will continue to support them on the path to membership, and will continue to support Ukraine's fight for freedom.