Question

Prove that if $A_1,A_2,\dots ,A_n$and $B_1,B_2,\dots ,B_n$ are sets such that $Aj\subseteq \text{Bifor}j=1,2,\dots ,n$, then$\underset{j=1}{\overset{n}{\cup }} Aj\subseteq \underset{j=1}{\overset{n}{\cup }} Bj$

Short answer

$\underset{j=1}{\overset{n}{\cup }} Aj\subseteq \underset{j=1}{\overset{n}{\cup }} Bj$

Step-by-step solution

Step: 1

If n=1,

$\begin{array}{r}\underset{j=1}{\overset{1}{\cup }} Aj\subseteq \underset{j=1}{\overset{1}{\cup }} Bj\\ A_1\subseteq B_1\end{array}$

it is true for n=1.

Step: 2

Let P(k) be true.

$\underset{j=1}{\overset{k}{\cup }} Aj\subseteq \underset{j=1}{\overset{k}{\cup }} Bj$

We need to prove that P(k+1) is true.

Step: 3

$\begin{array}{r}z\in \underset{j=1}{\overset{k+1}{\cup }} Aj\\ z\in \left(\underset{j=1}{\overset{k}{\cup }} Aj\right)\cup A_{k+1}\\ lly,z\in \left(\underset{j=1}{\overset{k}{\cup }} Bj\right)\cup B_{k+1}\\ z\in \underset{j=1}{\overset{k}{\cup }} Bj\end{array}$

It is true for P(k+1) is true.