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Q39E

Expert-verifiedFound in: Page 330

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 $\underset{j=1}{\overset{n}{\cap}}\u200aAj\subseteq \underset{j=1}{\overset{n}{\cap}}\u200aBj$

$\underset{j=1}{\overset{n}{\cap}}\u200aAj\subseteq \underset{j=1}{\overset{n}{\cap}}\u200aBj$

If n=1,

$\begin{array}{r}\underset{j=1}{\overset{1}{\cap}}\u200aAj\subseteq \underset{j=1}{\overset{1}{\cap}}\u200aBj\\ {A}_{1}\subseteq {B}_{1}\end{array}$

it is true for n=1.

Let P(k) be true.

$\underset{j=1}{\overset{k}{\cap}}\u200aAj\subseteq \underset{j=1}{\overset{k}{\cap}}\u200aBj$

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

$\begin{array}{r}z\in \underset{j=1}{\overset{k+1}{\cap}}\u200aAj\\ z\in \left(\underset{j=1}{\overset{k}{\cap}}\u200aAj\right)\cap {A}_{k+1}\\ lly,z\in \left(\underset{j=1}{\overset{k}{\cap}}\u200aBj\right)\cap {B}_{k+1}\\ z\in \underset{j=1}{\overset{k}{\cap}}\u200aBj\end{array}$

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

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