q)
is true, we assume that p is true and use that to show that q must also be true.2. The question can be rewritten as an if--then statement. If r and
s are two rational numbers, then the product r·s must be a rational number. To prove that
an if--then statement (p q)
is true, we assume that p is true and use that to show that q must also be true.
Proof Suppose r and s are rational
numbers. Then by the definition of rational, r = a/b and s = c/d for some integers a, b, c and d, where b 
0 and d
0
| Multiplying the two expressions together: r · s | = |
a/b · c/d |
= |
ac/bd |
Let x = ac and y = bd. Since Z is closed under
multiplication, both x and y are integers. Thus the product r · s = x/y where x, y 
Z and y 
0 (since b,d 0). Therefore the product of any
two rational numbers is rational.