Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Prove that if $A_{1}, A_{2}, \dots, A_{n}$ and $B_{1}, B_{2}, \dots, B_{n}$ are sets such that $A j \subseteq Bjforj = 1, 2, \dots, n$ , then $\cap_{j = 1}^{n} A j \subseteq \cap_{j = 1}^{n} B j$

$\cap_{j = 1}^{n} A j \subseteq \cap_{j = 1}^{n} B j$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step: 1

If n=1,

$\begin{aligned} \cap_{j = 1}^{1} A j \subseteq \cap_{j = 1}^{1} B j \\ A_{1} \subseteq B_{1} \end{aligned}$

it is true for n=1.

Step: 2

Let P(k) be true.

$\cap_{j = 1}^{k} A j \subseteq \cap_{j = 1}^{k} B j$

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

Step: 3

$\begin{aligned} z \in \cap_{j = 1}^{k + 1} A j \\ z \in (\cap_{j = 1}^{k} A j) \cap A_{k + 1} \\ l l y, z \in (\cap_{j = 1}^{k} B j) \cap B_{k + 1} \\ z \in \cap_{j = 1}^{k} B j \end{aligned}$

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

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Discrete Mathematics and its Applications

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Short Answer

Prove that if $A_{1}, A_{2}, \dots, A_{n}$ and $B_{1}, B_{2}, \dots, B_{n}$ are sets such that $A j \subseteq Bjforj = 1, 2, \dots, n$ , then $\cap_{j = 1}^{n} A j \subseteq \cap_{j = 1}^{n} B j$

Step by Step Solution

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Prove that if $A_{1}, A_{2}, \dots, A_{n}$ and $B_{1}, B_{2}, \dots, B_{n}$ are sets such that $A j \subseteq Bjforj = 1, 2, \dots, n$ , then $\cap_{j = 1}^{n} A j \subseteq \cap_{j = 1}^{n} B j$