Algebra of Sets

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

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

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

According to Idempotent property of intersection of sets, for any set $P$, $P\cap P$ is 

Which of the following is not equal to set $A$?

Which of the following is equal to $A\cup A\text{'}$?

Let $A=\left\{1, 2, 4, 5\right\}, B=\left\{2, 3, 5, 6\right\}, C=\left\{4, 5, 6, 7\right\}$. Verify the following property of intersection:

$\left(A\cap B\right)\cap C=A\cap \left(B\cap C\right)$

Let $A=\left\{1, 2, 4, 5\right\}, B=\left\{2, 3, 5, 6\right\}$. Verify the following identity:

$A\cap B=B\cap A$

By giving one example show that Intersection operation in sets follows commutative law.

$A=\left\{2,5\right\}$, $B=\left\{5,8\right\}$ and $C=\left\{4,6\right\}$. Prove the associative property of union of sets.

$A=\left\{8,9\right\}$, $B=\left\{2,3\right\}$ and $C=\left\{6,7\right\}$. Prove the associative property of union of sets.

$A=\left\{3,4\right\}$, $B=\left\{4,5\right\}$ and $C=\left\{6,7\right\}$. Prove the associative property of union of sets.

$A=\left\{1,2\right\}$, $B=\left\{2,3\right\}$ and $C=\left\{4,5\right\}$. Prove the associative property of union of sets.

$A=\left\{1,2,3\right\}$ and $B=\left\{2,3,4,5\right\}$. Prove the commutative property of union of two sets with the given sets.

$A=\left\{6,2,5,3\right\}$, $B=\left\{1,2,6,4\right\}$ and $C=\left\{2,3,5,1\right\}$. Prove the associative property of sets intersection.

Prove the associative property of sets intersection on $A=\left\{5,6,7,8\right\}$, $B=\left\{1,2,8,5\right\}$ and $C=\left\{2,4,5,6\right\}$.

$A=\left\{1,2,3,4\right\}$, $B=\left\{1,3,4,5\right\}$ and $C=\left\{2,4,5,6\right\}$. Prove the associative property of sets intersection.

If $A=\left\{\mathrm{prime} \mathrm{numbers} \mathrm{under} 25\right\}$ and $\mathrm{B}=\left\{\mathrm{odd} \mathrm{numbers} \mathrm{under} 25\right\}$. Prove that, $A\cap B$ belongs to set of numbers under $25$.

If $A=\left\{\mathrm{prime} \mathrm{numbers} \mathrm{under} 15\right\}$ and $\mathrm{B}=\left\{\mathrm{odd} \mathrm{numbers} \mathrm{under} 15\right\}$. Prove that, $A\cap B$ belongs to set of numbers under $15$.

If $A=\left\{\mathrm{prime} \mathrm{numbers} \mathrm{under} 20\right\}$ and $\mathrm{B}=\left\{\mathrm{odd} \mathrm{numbers} \mathrm{under} 20\right\}$. Prove that, $A\cap B$ belongs to set of numbers under $20$.

If $A=\left\{\mathrm{prime} \mathrm{numbers} \mathrm{under} 10\right\}$ and $\mathrm{B}=\left\{\mathrm{odd} \mathrm{numbers} \mathrm{under} 10\right\}$. Prove that, $A\cap B$ belongs to set of numbers under $10$.

If $A=\left\{2,4,6,8,10,12,14\right\}$ and $B=\left\{2,3,5,7,11,13\right\}$ and $U=\left\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\right\}$, then prove that, $A\cap B\ne U$.