Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)
Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)
3rd Edition
ISBN: 9780134689555
Author: Edgar Goodaire, Michael Parmenter
Publisher: PEARSON
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Chapter 6, Problem 1RE

Suppose A and B are nonempty finite sets and | A B | < | A | + | B | . Show that A B .

Expert Solution & Answer
Check Mark
To determine

To prove: AB.

Explanation of Solution

Given Information:

Suppose A and B are nonempty finite sets and |AB|<|A|+|B|.

Explanation:

Proof:

By Principle of Inclusion-Exclusion,

A finite set, A1,A2,..........,An, the number of element in the union A1A2.....An is,

|A1A2.....An|=i|Ai|i<j|AiAj|+i<j<k|AiAjAk|...+(1)n+1|A1A2....An|

Where the first sum is over all i, the second is over all pairs i, j with i<j, the third sum is over all triples i, j, k with i<j<k, and so forth.

By the Principle of Inclusion-Exclusion we have,

|AB|=|A|+|B||AB||AB|=|A|+|B||AB|

Since,

|AB|<|A|+|B|

Thus,

|A|+|B||AB|>0

Therefore,

|AB|>0

Hence,

ABϕ

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Chapter 6 Solutions

Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)

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