R D Sharma Solutions for Chapter: Sets, Exercise 6: EXERCISE 1.6

Let $A=\{1, 2, 4, 5\}, B=\{2, 3, 5, 6\}, C=\{4, 5, 6, 7\}$ Verify the following identity:

$A\cap (B-C)=(A\cap B)-(A\cap C)$

Using properties of sets, show that for any two sets $A \mathrm{and} B, (A\cup B)\cap (A\cup B^{\text{'}})=A.$

For any two sets of$A \mathrm{and} B$, prove that: $A^{\text{'}}\cup B=U\Rightarrow A\subset B$

For any two sets of $A$ and $B$, prove that: $B^{\text{'}}\subset A^{\text{'}}\Rightarrow A\subset B$

Is it true that for any set $A$ and $B$,  $P \left(A\right)\cup P \left(B\right)=P (A\cup B)?$ Justify your answer.

Show that for any sets $A \mathrm{and} B,$ $A=(A\cap B)\cup (A-B)$

Show that for any sets $A \mathrm{and} B,$ $A\cup (B–A)=A\cup B$.

Each set $X$, contains $5$ elements and each set $Y$, contains $2$ elements and $\underset{r=1}{\overset{20}{\cup }}{X}_r=S=\underset{r=1}{\overset{n}{\cup }}{Y}_r$ . If each element of $S$ belongs to exactly $10$ of the $X_r\text{'}s$ and to exactly $4$ of $Y_r\text{'}s$, then find the value of $n$.