If $A$ and $B$ are subsets of the universal set $U$, then show that
(i) $A \subset A \cup B$
(ii) $A \subset B \Leftrightarrow A \cup B=B$
(iii) $(A \cap B) \subset A$
$$\begin{array}{ll} \text {(i) Let } & x \in A \\ \Rightarrow & x \in A \text { or } x \in B \Rightarrow x \in A \cup B \\ \text { Hence, } & \subset A \cup B \end{array}$$
$$\begin{aligned} &\text { (ii) If }\\ &\begin{array}{lr} & A \subset B \\ \text { Let } & x \in A \cup B \end{array} \end{aligned}$$
$$\begin{aligned} &\begin{array}{lcr} \Rightarrow & x \in A \text { or } x \in B \Rightarrow x \in B & {[\because A \subset B]} \\ \Rightarrow & A \cup B \subset B & \ldots \text { (i) } \\ \text { But } & B \subset A \cup B & \ldots \text { (ii) } \end{array}\\ \end{aligned}$$
From Eqs. (i) and (ii),
$\begin{aligned} & A \cup B=B \\ & \text{If}\quad A \cup B=B\end{aligned}$
Let $y \in A$
$\Rightarrow \quad y \in A \cup B \Rightarrow y \in B \quad[\because A \cup B=B]$
$\begin{array}{ll}\Rightarrow & A \subset B \\ \text { Hence, } & A \subset B \quad \Leftrightarrow \quad A \cup B=B\end{array}$
(iii) Let $$x \in A \cap B$$
$\Rightarrow \quad x \in A$ and $x \in B \quad \Rightarrow \quad x \in A$
Hence, $$A \cap B \subset A$$
Given that $N=\{1,2,3, \ldots, 100\}$. Then, write
(i) the subset of $N$ whose elements are even numbers.
(ii) the subset of $N$ whose elements are perfect square numbers.
We have, $\quad N=\{1,2,3,4, \ldots ., 100\}$
(i) Required subset $=\{2,4,6,8, \ldots, 100\}$
(ii) Required subset $=\{1,4,9,16,25,36,49,64,81,100\}$
If $X=\{1,2,3\}$, if $n$ represents any member of $X$, write the following sets containing all numbers represented by
(i) $4 n$
(ii) $n+6$
(iii) $\frac{n}{2}$
(iv) $n-1$
Given, $$X=\{1,2,3\}$$
(i) $\{4 n \mid n \in X\}=\{4,8,12\}$
(ii) $\{n+6 \mid n \in X\}=\{7,8,9\}$
(iii) $\left\{\left.\frac{n}{2} \right\rvert\, n \in x\right\}=\left\{\frac{1}{2}, 1, \frac{3}{2}\right\}$
(iv) $\{n-1 \mid n \in X\}=\{0,1,2\}$
If $Y=\{1,2,3, \ldots, 10\}$ and $a$ represents any element of $Y$, write the following sets, containing all the elements satisfying the given conditions.
(i) $a \in Y$ but $a^2 \notin Y$
(ii) $a+1=6, a \in Y$
(iii) $a$ is less than 6 and $a \in Y$
Given, $$Y=\{1,2,3, \ldots, 10\}$$
(i) $\left\{a: a \in Y\right.$ and $\left.a^2 \notin Y\right\}=\{4,5,6,7,8,9,10\}$
(ii) $\{a: a+1=6, a \in Y\}=\{5\}$
(iii) is less than 6 and $a \in Y\}=\{1,2,3,4,5$, $\}$
$A, B$ and $C$ are subsets of universal set $U$. If $A=\{2,4,6,8,12,20\}$, $B=\{3,6,9,12,15\}, C=\{5,10,15,20\}$ and $U$ is the set of all whole numbers, draw a Venn diagram showing the relation of $U, A, B$ and $C$.
