   Chapter 2.2, Problem 23E

Chapter
Section
Textbook Problem

Let x and y be integers, and let m and n be positive integers. Use mathematical induction to prove the statements in Exercises 18 − 23 . ( The definitions of x n and n x are given before Theorem 2.5 in Section 2.1 )           m ( n x ) = ( m n ) x

To determine

To prove: m(nx)=(mn)x by using mathematical induction, where x and y be integers, and m and n are positive integers.

Explanation

Given information:

The given statement is, “ m(nx)=(mn)x.”

Formula used:

For all positive integers n, the statement Pn is true if,

a. Pn is true for n=1

b. The truth of Pk always implies that Pk+1 is true.

Proof:

Consider the statement, “ m(nx)=(mn)x ”where x and y be integers, and m and n are positive integers.

For m=1,n=1.

Taking L.H.S.,

1(1x)

By using 1 as identity of multiplication,

=x

Taking R.H.S,

(11)x

By using 1 as identity of multiplication,

=x

Hence, 1(1x)=(11)x

Thus, the statement is true for m=1,n=1.

Assume the statement is true for m=k,n=k'.

k(k'x)=(kk')x

For m=k+1,n=k'+1,

The left side is,

(k+1)((k'+1)x)

By using right distributive law,

=(k+1)(k'x+1x)

By using 1 as identity of multiplication,

=(k+1)(k&

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