Answer
a product of two positive real numbers is greater than 100, then at least one of the numbers is greater than 10. proving this statement by contrapositions.
Work Step by Step
If a product of two positive real numbers is greater than 100, then at least one of the numbers is greater than 10.
Proof (by contraposition):
-[To go by contraposition, we must prove that ∀ positive real numbers, r and s, if r ≤ 10 and s ≤ 10, then r s ≤ 100.]
-Suppose r and s are positive real numbers and r ≤ 10 and s ≤ 10.
-By the algebra of inequalities, rs ≤ 100.
-[To derive this fact, multiply both sides of r ≤ 10 by s to obtatin r s ≤ 10s.
- And multiply both sides of s ≤ 10 by 10 to obtain 10s ≤ 10·10 = 100.
- By transitivity of ≤, then, r s ≤ 100.] But this is what was to be shown.