Question

Consider the statement: The cube of any rational number is a rational number. a. Write the statement formally using a quantifier and a variable. b. Determine whether the statement is true or false and justify your answer.

Answer 1

Answer:

(a)∀ r∈ℝ, r∈ℚ, r³∈ℚ

(b)True

Step-by-step explanation:

Definition

1. A real number is said to be rational if and only if there exists two integers a and b such that $\frac{a}{b}=r$ where b≠0.

2. If the product of three numbers is zero, then at least one of the numbers must be zero.

Given Statement

The cube of any rational number is a rational number.

This can be rewritten as:

For all real numbers, if the number is rational then its cube is rational.

If we introduce our variable r,

For all real number r, if r is rational, then r³ is rational.

∀ r∈ℝ, r∈ℚ, r³∈ℚ

(b)The Statement is True.

To prove: The cube of any rational number is rational.

Proof:

We assume that r is a rational number and prove that its cube is a rational number.

By definition, there exists integers a and b such that:

$r=\frac{a}{b}$ where b≠0.

$r^3=(\frac{a}{b})^3$

$r^3=\frac{a^3}{b^3}$

Since the cube of an integer is also an integer, a³ and b³ are also integers.

Since b is non-zero, then the zero product property tells us that its cube is also non-zero.

Conclusion:

$r^3=\frac{a^3}{b^3}$ is rational since a³ and b³ are integers and b is non-zero.