Operations on Sets

Which of the following binary operations for set $N$ are associative and/or commutative:

$\left(a\right)$  $a*b=1 \forall a,b\in N$

$\left(b\right)$  $a*b=\frac{a+b}{2} \forall a,b\in N.$

$A=\left\{a,b\right\}$, $B=\left\{e,f\right\}$. Verify closure property of union of two sets.

$A=\left\{a,b,c\right\}$, $B=\left\{d,e,f\right\}$. Verify closure property of union of two sets.

$A=\left\{1,2,3\right\}$, $B=\left\{2,3,4,5\right\}$. Verify closure property of union of two sets.

$A=\left\{1,2,3,4\right\}$, $B=\left\{2,3,4,5,6\right\}$. Verify closure property of union of two sets.

$A=\left\{1,2\right\}$, $B=\left\{2,3\right\}$. Verify closure property of union of two sets.

$A=\left\{2,3,5\right\}$. Prove idempotent property of union of sets for given set.

$A=\left\{m,n\right\}$. Prove idempotent property of union of sets for given set.

$A=\left\{a,b\right\}$. Prove idempotent property of union of sets for given set.

$A=\left\{3,4\right\}$. Prove idempotent property of intersection of sets for given set.

$A=\left\{1,2\right\}$. Prove idempotent property of union of sets for given set.

$A=\left\{1,2,3\right\}$ and $U=\left\{1,2,3,4\right\}$. Find $A\cap U$.

$A=\left\{1,4\right\}$ and $U=\left\{1,2,3,4\right\}$. Find $A\cap U$.

$A=\left\{2,4\right\}$ and $U=\left\{1,2,3,4\right\}$. Find $A\cap U$.

$A=\left\{1,3\right\}$ and $U=\left\{1,2,3,4\right\}$. Find $A\cap U$.

$A=\left\{1,2\right\}$ and $U=\left\{1,2,3,4\right\}$. Find $A\cap U$.

$A=\left\{1,2,3\right\}$. Find $A\cup \varnothing$.

$A=\left\{x,y\right\}$. Find $A\cup \varnothing$.

$A=\left\{a,b\right\}$. Find $A\cup \varnothing$.

$A=\left\{3,4\right\}$. Find $A\cup \varnothing$.