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D Theorem: If rand s are rational numbers, then the product of rands is a rational number. Which facts are assumed in a direct proof of the theorem? Or-a/b, and sc/d, where a, b, c, d are integers and a 0 and c-0. Ora/b, and sc/d, where a, b, c, d are integers and b*0 and d 0. Ors= a/b, where a and b are integers a 0. Ors= a/b, where a and b are integers b 0. Question 28 Theorem: For any two real numbers, x and y, if x and y are both rational then x + y is also rational. Which facts are assumed and which facts are proven in a direct proof of the theorem? O Assumed: x + y is rational Proven: x is rational or y is rational O Assumed: x + y is irrational Proven: x is irrational and y is irrational O Assumed: x is rational and y is rational Proven: x + y is rational O Assumed: x is rational or y is rational Question 29 Theorem: For any two real numbers, x and y, if x and y are both rational then x + y is also rational. Which facts are assumed and which facts are proven in a proof by contrapositive of the theorem? Assumed: x + y is irrational Proven: x is irrational or y is irrational O Assumed: x is rational and y is rational Proven: x + y is rational O Assumed: x + y is irrational Proven: x is irrational and y is irrational O Assumed: x is rational or y is rational Proven: x + y is rational 3 p
Theorem: For any real number x, if x2 - 6x +5> 5, then x 25 or xs 1. Which facts are assumed and which facts are proven in a proof by contrapositive of the theorem? Assumed: x < 5 orx > 1 Proven: x2 - 6x + 5 ≤ 5 Assumed: 1 < x < 5 Proven: x² - 6x +5 ≤5 Assumed: x ≥ 5 or x ≤ 1 Proven: x2 - 6x + 5 ≤ 5 O Assumed: x ≥ 5 and x ≤1 Proven: x2 - 6x + 5 ≤ 5 3 ptm
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