> 
> 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?](https://math.stackexchange.com/questions/110760/wedge-cap-and-vee-cup-between-logic-and-set-theory-always-interchangeabl)
> 
> 
> 

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?