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CH2 (2010)

# CH2 (2010) - Discrete Mathematics 1st Edition Kevin Ferland...

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Discrete Mathematics, 1st Edition Kevin Ferland Chapter 2 Basic Proof Writing 1

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2.1 Direct Demonstration 2 Ch2-p70 Show: all of the points (−4, −5), (2, −2), and (8, 1) lie on the common line L .
EXAMPLE 2.1 -- Proof Let L be the line given by the equation y = 3. Observe that and Therefore, all of the points (−4, −5), (2, −2), and (8, 1) lie on the common line L . 3 Ch2-p70

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EXAMPLE 2.3 Show: There exist sets A and B such that | A B | < | A | + | B |. 4 Ch2-p71
EXAMPLE 2.3 -- Proof Let A = {1, 2} and B = {2, 3}. So A B = {1, 2, 3}. Observe that | A B | = 3 < 2 + 2 = | A | + | B | . 5 Ch2-p71

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EXAMPLE 2.5 Prove or Disprove: x R, if x < 2, then x 2 < 4. 6 Ch2-p71
EXAMPLE 2.5 ( Solution ) Counterexample Let x = −3. So x 2 = 9. Observe that x < 2 and x 2 ≥ 4. That is, for x = −3, it is not true that if x < 2, then x 2 < 4 . 7 Ch2-p72

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2.2 and 2.3 General Demonstration Often, we want to prove a statement of the form x U , p ( x ) . 8 Ch2-p75
EXAMPLE 2.8 Show: x R, x 2 + 1 > 0. 9 Ch2-p75

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EXAMPLE 2.8 -- Proof Let a R be arbitray. Since the square of any real number is nonnegative, we have a 2 ≥ 0.
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