   Chapter 9.3, Problem 45ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Use mathematical induction to prove Theorem 9.3.1.

To determine

To prove:

N(A)=N(A1)+N(A2)++N(Ak) by mathematical induction.

Explanation

Given information:

Suppose a finite set A equals of k distinct mutually disjoint subsets A1,A2,Ak.

Proof:

PROOF BY INDUCTION:

Let P(n) be “If a finite set A is the union of n distinct mutually disjoint subsets A1,A2,...,An, then: N(A)=N(A1)+N(A2)+...+N(An)

Basis step: n = 1

Let A be the union of 1 distinct mutually disjoint subset A1

A=i=11Ai=A1

Since A = A1, the two sets need to contain the same number of elements:

N(A)=N(A1)

Thus P (1) is true.

Inductive step:

Let P(k) be true, thus if A is the union of k distinct mutually disjoint subsets A1,A2,...,Ak, then: N(A)=N(A1)+N(A2)+...+N(Ak)

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

Let A be the union of k + 1 distinct mutually disjoint subsets A1,A2,...,Ak,Ak+1

A=i=1k+1Ai

Let B=i=1kAi, thus B is the union of the first k distinct mutually disjoint sets. Since P(k) is true:

N(B)=N(A1)+N(A2)+..

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