Exercise 2.4.2: Proving statements about rational numbers with direct proofs. About Prove each of the following statements using a direct proof. (a) The product of two rational numbers is a rational number. Solution (b) The quotient of a rational number and a non-zero rational number is a rational number. Solution (c) If x and y are rational numbers then is also a rational number.

See Explanation

Step-by-step explanation:

(a) Proof: Product of two rational numbers

Using direct proofs.

Let the two rational numbers be A and B.

Such that:

The product:

Proved, because 1/3 is rational

(b) Proof: Quotient of a rational number and a non-zero rational number

Using direct proofs.

Let the two rational numbers be A and B.

Such that:

The quotient:

Express as product

Proved, because 3/4 is rational

(c) x + y is rational (missing from the question)

Using direct proofs.

Let x and y be

Such that:

The sum:

Take LCM

Proved, because 7/6 is rational

The above proof works for all values of A, B, x and y; as long as they are rational values

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