09.04.2023

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.

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(a) Proof: Product of two rational numbers

Using direct proofs.

Let the two rational numbers be A and B.

Such that:

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

The product:

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

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:

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

The quotient:

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

Express as product

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

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:

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

The sum:

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

Take LCM

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

Exercise 2.4.2: Proving statements about rational, №17886909, 09.04.2023 20:39

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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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:

A = \frac{1}{2}

B = \frac{2}{3}

The product:

A * B = \frac{1}{2} * \frac{2}{3}

A * B = \frac{1}{1} * \frac{1}{3}

A * B = 1 * \frac{1}{3}

A * B = \frac{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 = \frac{1}{2}

B = \frac{2}{3}

The quotient:

A / B = \frac{1}{2} / \frac{2}{3}

Express as product

A / B = \frac{1}{2} / \frac{3}{2}

A / B = \frac{1*3}{2*2}

A / B = \frac{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 = \frac{1}{2}

y = \frac{2}{3}

The sum:

x + y = \frac{1}{2} + \frac{2}{3}

Take LCM

x + y = \frac{3+4}{6}

x + y = \frac{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

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