Algebra of Sets

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

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

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

An organization awarded $48$ medals in event ${}^{\text{'}}A^{\text{'}}$, $25$ in event ${}^{\text{'}}B^{\text{'}}$ and $18$ in event ${}^{\text{'}}C^{\text{'}}$. If these medals went to total $60$ men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?

When $A\subset B$, then $A-B=$

Let $S=\left\{4,6,9\right\}$ and $T=\left\{9,10,11,....,1000\right\}$. If $A=\left\{a_1+a_2+a_3+...+a_k, k\in N, a_1, a_2, a_3,...., a_k\in S\right\}$, then the sum of elements in the set $T-A$ is equal to

If $A=\left\{x:\frac{x}{2}\in Z, 0\le x\le 10\right\}$, $B=\left\{x: x \text{is one digit prime}\right\}$ and $C=\left\{x:\frac{x}{3}\in N, x\le 12\right\}$.

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

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.

$A=\left\{a,b\right\}$, $B=\left\{b,c\right\}$ and $C=\left\{d,e\right\}$. Prove the associative property of union of sets.

$A=\left\{2,5\right\}$, $B=\left\{5,8\right\}$ 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.

$A=\left\{4,5,6\right\}$ and $B=\left\{5,6,7\right\}$. Prove the commutative property of union of two sets with the given 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.

If $A=\left\{\mathrm{prime} \mathrm{numbers} \mathrm{from} 10 \mathrm{to} 25\right\}$ and $B=\left\{\mathrm{odd} \mathrm{numbers} \mathrm{from} 10 \mathrm{to} 25\right\}$. Prove that, $A\cap B$ belongs to set of numbers under $25$.

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 $10$.