Determine the truth value of each statement

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Reference no: EM131131229

Solve the following three problems. Write your solutions in English.

1. We will derive the quadratic equation by solving a system of linear equations. The equation ax2 + bx + c = 0 with a ≠ 0 has real solutions r, s if and only if ax2 + bx + c = a(x - r)(x - s). The calculation below shows that the factorization exists if and only if b2 - 4ac ≥ 0.

a) By equating coefficients of corresponding powers of x, obtain the equations r + s = -b/a and rs = c/a. Use these to prove that (r - s)2 = (b2 - 4ac)/a2.

b) From (a), obtain r + s = -b/a and r - s = √(b2 - 4ac)/a. Solve this system for r, s in terms of a, b, c.

c) What happens if the second equation in (b) is r - s = -√(b2 - 4ac)/a?

2. Given a real number x, let A be the statement "1/2 < x < 5/2", let B be the statement x ∈ Z, let C be the statement x2 = 1, and let D be the statement "x = 2". Which statements below are true for all x ∈ R?

a) A C.

b) B C.

c) (A ∧ B) C.

d) (A ∧ B) (C ∨ D).

e) C (A ∧ B).

f) D [A ∧ B ∧ (¬C)]. g) (A ∨ C) B.

3. Let x, y be integers. Determine the truth value of each statement below.

a) xy is odd if and only if x and y are odd.

b) xy is even if and only if x and y are even.

Reference no: EM131131229

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