Algebra of Sets

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

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

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

Let $A$ and $B$ be two sets such that $n\left(A-B\right)=14+x, n\left(B-A\right)=3x$ and $n\left(A\cap B\right)=x$. Draw a venn diagram to illustrate this information. If $n\left(A\right)=n\left(B\right)$, then find the value of $x$.

Let $A$ be the set of non-negative integers, $I$ is the set of integers, $B$ is the set of non-positive integers, $E$ is the set of even integers and $P$ is the set of prime numbers, then

In a study about a pandemic, data of $900$ persons was collected. It was found that

$190$ persons had symptom of fever,

$220$ persons had symptom of cough,

$220$ persons had symptom of breathing problem,

$330$ persons had symptom of fever or cough or both,

$350$ persons had symptom of cough or breathing problem or both,

$340$ persons had symptom of fever or breathing problem or both,

$30$ persons had all three symptoms (fever, cough and breathing problem).

If a person is chosen randomly from these $900$ persons, then the probability that the person has at most one symptom is ______.

$A=\left\{a,b\right\}$, $B=\left\{e,f\right\}$. 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 $A\cup \varnothing$.