Algebra of Sets

In a class, $20$ students opted for Physics, $17$ for Maths, $5$ for both and $10$ for other subjects. how many students are there in the class?

A survey of $100$ persons revealed that $72$ of them had eaten at restaurant P and that $52$ of them had eaten at restaurant Q. Which of the following could not be the number of persons in the surveyed group who had eaten at both P and Q?

Study the following information carefully and answer the given questions:

A committee prepared a list of twenty players each in Test, ODI and T$20$ cricket tournaments, out of $39$ players. Three players are there who are in Test and ODI list but not in T$20$. Four players are there who are in Test and T$20$ list but not in ODI, and same number of players are there who are in all the three lists.

How many players are there who are in exactly one list?

In a community of $175$ persons, $40$ read the Times, $50$ read the Samachar and $100$ do not read any. How many persons read both the papers ?

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

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

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

Which of the following is not correct?

Intersection operation in sets follows associative law.

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.

In a class there are $50$ students and each student has a hobby as painting, dancing, singing or writing. $24$ students have their hobby as painting or singing, $10$ students have their hobby as painting,$7$students have their hobby as both painting and singing whereas$10$ students like to dance and $8$ students like both dancing and writing. Each student has atmost two hobbies. Find the number of students who like singing or only writing or no hobby at all :

$A=\left\{a,b\right\}$, $B=\left\{b,c\right\}$ 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 $A=\left\{2,4,6,8,10,12,14\right\}$ 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, $A\cap B\ne U$.

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