Take the 2-minute tour ×  Update : Should have left the
Arithmetic out of this question, the new modified question is
posted here : $\wedge,\cap$ and $\vee,\cup$ between Logic and
Set Theory always interchangeable?  Although have not seen this
anywhere stated, but so often mentally swapping the frameworks
of Logic, Set Theory and Arithmetic following operation
$\wedge,\cap,\times$ and $\vee,\cup,+$ are seem to be always
interchangeable (Isomorphic?).  My question is are
$\wedge,\cap,\times$ the same thing and just different symbols
are being used depending on the framework? same question
regarding $\vee,\cup,+$  PS: In arithmetic setting we get the
case of $[0, \text{any number other than 0}] \equiv [false ,
true] $  For example De Morgan's laws for Sets and Logic just
becomes the distributive and associate laws.  What about
infinite cases? does this type of intuition break down between
Arithmetic, Logic and Set theory?  share|improve this question
In Set Theory, $\cap$ and $\cup$ distribute over each other: