Prove the following:

$A\cup B=U$ and $A\cap B=\varphi \Rightarrow B=A^{\text{'}}$, where $U$ is the universal set

Prove the following:

$A\cup B=A$ for all $A\Rightarrow B=\varphi$

Prove that $A-\underset{i=i}{\overset{n}{\cup }}{B}_i=\underset{i=1}{\overset{n}{\cap }}(A-B_i)$ and $A-\underset{i=1}{\overset{n}{\cap }}{B}_i=\underset{i=1}{\overset{n}{\cup }}(A-B_i)$.

Prove that

$|A\cup B\cup C|=\left|A\right|+\left|B\right|+\left|C\right|+|A\cap B\cap C|-|A\cap B|-|B\cap C|-|C\cap A\mid$

If $X$ and $Y$ are two sets such that $X\cup Y$ has $20$ objects, $X$ has $10$ objects and $Y$ has $15$ objects; how many objects does $X\cap Y$ have ?

In a group of $450$ people, $300$ can speak Hindi and $250$ can speak English. How many people can speak both Hindi and English ?

In a group of people, $37$ like coffee, $52$ like tea and each person in the group likes at least one of the two drinks. $19$ people like both tea and coffee, how many people are in the group?

In a class of $35$ students, each student likes to play either cricket or hockey. $24$ students like to play cricket and $5$ students like to play both the games; how many students play hockey?

In a class of $400$ students, $100$ were listed as taking apple juice, $150$ as taking orange juice and $75$ were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice nor orange juice?