In "[$\wedge,\cap,\times$ and $\vee,\cup,+$ are always interchangeable?](https://math.stackexchange.com/questions/110744/wedge-cap-times-and-vee-cup-are-always-interchangeable)" 
It has been shown that arithmetic shouldn't be included. So the new modified question is: 

The analogy of $\wedge,\cap$ and $\vee,\cup$ are seem to be in Logic and Set Theory are seem to be so similar that one might might be forgiven to think they are the same thing but with different notation.

My question is are $\wedge,\cap$ the same thing and just different symbols are being used depending on the framework? same question regarding $\vee,\cup$

What about infinite cases? does this type of intuition break down between Logic and Set theory?