Answer:
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:
[tex]A = \frac{1}{2}[/tex]
[tex]B = \frac{2}{3}[/tex]
The product:
[tex]A * B = \frac{1}{2} * \frac{2}{3}[/tex]
[tex]A * B = \frac{1}{1} * \frac{1}{3}[/tex]
[tex]A * B = 1 * \frac{1}{3}[/tex]
[tex]A * B = \frac{1}{3}[/tex]
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:
[tex]A = \frac{1}{2}[/tex]
[tex]B = \frac{2}{3}[/tex]
The quotient:
[tex]A / B = \frac{1}{2} / \frac{2}{3}[/tex]
Express as product
[tex]A / B = \frac{1}{2} / \frac{3}{2}[/tex]
[tex]A / B = \frac{1*3}{2*2}[/tex]
[tex]A / B = \frac{3}{4}[/tex]
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:
[tex]x = \frac{1}{2}[/tex]
[tex]y = \frac{2}{3}[/tex]
The sum:
[tex]x + y = \frac{1}{2} + \frac{2}{3}[/tex]
Take LCM
[tex]x + y = \frac{3+4}{6}[/tex]
[tex]x + y = \frac{7}{6}[/tex]
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
