Does (A∩C)=(B∩C)∧(A∪C)=(B∪C) mean {(A∩C)=(B∩C)}∧{(A∪C)=(B∪C)} ?

Question

I don't know about set operator's priority exactly. I think '=' operator is prior to '^'. Is that right?

Answer 1

Yes:

The $\land$ symbol goes between claims and produces a new claim.

The $=$ symbol goes between things and produces a claim. (Here the "things" appear to be sets).

$B\cap C$ and $A\cup C$ are both things (i.e. sets), so the $\land$ symbol cannot connect them. The operand on each side of the $\land$ has to include the $=$ signs, because that's the only thing in the formula that produces claims.