Chapter 1.1, Problem 55E

Solution

To state: Whether the statement, a product of real numbers is zero if and only if every factor in the product is zero, is true or false.

False.

Given Information:

A product of real numbers is zero if and only if every factor in the product is zero

Explanation:

The given statement is false.

Consider the real numbers $x_1,x_2,x_3,\mathrm{...},x_n$ , $n>1$ .

The product of real numbers $x_1,x_2,x_3,\mathrm{...},x_n$ is $x_1{x}_2{x}_3\mathrm{....}{x}_{n-1}{x}_n$ .

Now, if any of the factors in $x_1{x}_2{x}_3\mathrm{....}{x}_{n-1}{x}_n$ is 0, then the entire product, that is, $x_1{x}_2{x}_3\mathrm{....}{x}_{n-1}{x}_n=0$ .

Thus, $x_1{x}_2{x}_3\mathrm{....}{x}_{n-1}{x}_n=0$ if there exist any $x_i=0$ , $1\le i\le n$ .

Hence, for a product of real numbers to be 0, it is not necessary that all the factors must be equal to 0.

Therefore, the given statement is false.