Question

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.

Answer 1

See Explanation

Explanation:

(a) Proof: Product of two rational numbers

Using direct proofs.

Let the two rational numbers be A and B.

Such that:

`A = (1)/(2)`

`B = (2)/(3)`

The product:

`A * B = (1)/(2) * (2)/(3)`

`A * B = (1)/(1) * (1)/(3)`

`A * B = 1 * (1)/(3)`

`A * B = (1)/(3)`

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:

`A = (1)/(2)`

`B = (2)/(3)`

The quotient:

`A / B = (1)/(2) / (2)/(3)`

Express as product

`A / B = (1)/(2) / (3)/(2)`

`A / B = (1*3)/(2*2)`

`A / B = (3)/(4)`

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:

`x = (1)/(2)`

`y = (2)/(3)`

The sum:

`x + y = (1)/(2) + (2)/(3)`

Take LCM

`x + y = (3+4)/(6)`

`x + y = (7)/(6)`

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