week 3 Discrete math

Theorem: If x is a positive integer less than 4, then (x+1)3 ≥ 4x . Which set of facts must be proven in a proof by exhaustion of the theorem? 3^3 ≥ 4^2 4^3 ≥ 4^3 1^3 ≥ 4^0 2^3 ≥ 4^1 3^3 ≥ 4^2 4^3 ≥ 4^3 2^3 ≥ 4^1 3^3 ≥ 4^2 4^3 ≥ 4^3 5^3 ≥ 4^4 2^3 ≥ 4^1 3^3 ≥ 4^2 4^3 ≥ 4^3

2^3 ≥ 4^1 3^3 ≥ 4^2 4^3 ≥ 4^3

Theorem: For any real number x, if 0 ≤ x ≤3 , then 15 - 8x + x2 > 0.Which facts are assumed and which facts are proven in a direct proof of the theorem? Assumed: 0 ≤ x and x ≤ 3 Proven: 15 - 8x + x2 > 0 Assumed: 0 ≤ x or x ≤ 3 Proven: 15 - 8x + x2 > 0 Assumed: 15 - 8x + x2 ≤ 0 Proven: 0 > x and x > 3 Assumed: 15 - 8x + x2 ≤ 0 Proven: 0 > x or x > 3

Assumed: 0 ≤ x and x ≤ 3 Proven: 15 - 8x + x2 > 0

Theorem: For any real number x , if x2 - 6x + 5 > 5 , then x ≥ 5 or x ≤ 1.Which facts are assumed and which facts are proven in a proof by contrapositive of the theorem? Assumed: x ≥ 5 or x ≤ 1 Proven: x2 - 6x + 5 ≤ 5 Assumed: x < 5 or x > 1 Proven: x2 - 6x + 5 ≤ 5 Assumed: x ≥ 5 and x ≤ 1 Proven: x2 - 6x + 5 ≤ 5 Assumed: 1 < x < 5 Proven: x2 - 6x + 5 ≤ 5

Assumed: 1 < x < 5 Proven: x2 - 6x + 5 ≤ 5

Theorem: For any real number x, if 0 ≤ x ≤ 3 , then 15 - 8x + x2 > 0. Which facts are assumed and which facts are proven in a proof by contrapositive of the theorem? Assumed: 15 - 8x + x2 ≤ 0 Proven: x < 0 or x > 3 Assumed: 0 ≤ x and x ≤ 3 Proven: 15 - 8x + x2 > 0 Assumed: 0 ≤ x or x ≤ 3 Proven: 15 - 8x + x2 > 0 Assumed: 15 - 8x + x2 ≤ 0 Proven: 0 < x and x > 3

Assumed: 15 - 8x + x2 ≤ 0 Proven: x < 0 or x > 3

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 Assumed: x is rational and y is rational Proven: x + y is rational Assumed: x is rational or y is rational Proven: x + y is rational Assumed: x + y is irrational Proven: x is irrational and y is irrational

Assumed: x + y is irrational Proven: x is irrational or y is irrational

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? Assumed: x + y is rational Proven: x is rational or y is rational Assumed: x is rational and y is rational Proven: x + y is rational Assumed: x is rational or y is rational Proven: x + y is rational Assumed: x + y is irrational Proven: x is irrational and y is irrational

Assumed: x is rational and y is rational Proven: x + y is rational

Select the mistake that is made in the proof given below. Theorem. The product of an even integer and any other integer is even. Proof. Suppose that x is an even integer and y is an arbitrary integer. Since x is even, x = 2k for some integer k. Therefore, xy = 2m for some integer m, which means that xy is even. Assuming facts that have not yet been proven. Generalizing from examples. Failure to properly introduce a variable. Misuse of existential instantiation.

Assuming facts that have not yet been proven.

Theorem: For any real number x, x + |x-5| ≥ 5.In a proof by cases of the theorem, there are two cases. One of the cases is that x > 5. What is the other case? x ≤ 5 x < 0 x < 5 x ≤ 0

x ≤ 5

Theorem: For any two real numbers, x and y, such that y ≠ 0, |x/y| = |x|/|y|. One of the cases in the proof of the theorem says: Since |x| = x and |y| = -y, |x|/|y| = x/(-y) = -x/y = |x/y|. Select the case that corresponds to this argument. x ≤ 0 and y > 0 x ≥ 0 and y < 0 x ≥ 0 and y > 0 x ≤ 0 and y < 0

x ≥ 0 and y < 0

Theorem: For any real numbers, x and y, max(x, y) = (1/2)(x + y + |x-y|).One of the cases in the proof of the theorem uses the assumptions that |x - y| = x - y. Select the case that corresponds to this argument. < y x ≥ 0 x ≥ y x < 0

x ≥ y