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Found in: Page 330

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# Prove that if ${A}_{1},{A}_{2},\dots ,{A}_{n}$ and ${B}_{1},{B}_{2},\dots ,{B}_{n}$ are sets such that $Aj\subseteq \text{Bjforj}=1,2,\dots ,n$ , then $\underset{j=1}{\overset{n}{\cap }} Aj\subseteq \underset{j=1}{\overset{n}{\cap }} Bj$

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

See the step by step solution

## Step: 1

If n=1,

$\begin{array}{r}\underset{j=1}{\overset{1}{\cap }} Aj\subseteq \underset{j=1}{\overset{1}{\cap }} Bj\\ {A}_{1}\subseteq {B}_{1}\end{array}$

it is true for n=1.

## Step: 2

Let P(k) be true.

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

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

## Step: 3

$\begin{array}{r}z\in \underset{j=1}{\overset{k+1}{\cap }} Aj\\ z\in \left(\underset{j=1}{\overset{k}{\cap }} Aj\right)\cap {A}_{k+1}\\ lly,z\in \left(\underset{j=1}{\overset{k}{\cap }} Bj\right)\cap {B}_{k+1}\\ z\in \underset{j=1}{\overset{k}{\cap }} Bj\end{array}$

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