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 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:
where b≠0.
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:
is rational since a³ and b³ are integers and b is non-zero.