CH1 (2010)_0304

# CH1 (2010)_0304 - Discrete Mathematics 1st Edition Kevin...

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Discrete Mathematics, 1st Edition Kevin Ferland Chapter 1 Logic and Sets 1

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1.1 Statement Forms and Logical Equivalences DEFINITION 1.1 A statement is a sentence that is either true or false, but not both. 2 Ch1-p9
EXAMPLE 1.1 The following sentences are statements. (a) 2 + 2 = 4. (b) 2 + 2 ≠ 4. (c) (d) The sine function is periodic and is an integer. (e) 10 2 > 2 10 or 2 10 > 10 2 . (f ) If e > 2, then e 2 > 4. (g) is a real number. 3 Ch1-p9-10

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EXAMPLE 1.2 The following sentences are not statements. (a) What is the sum 2 + 2? (b) Evaluate the sum 2 + 2. (c) This sentence is false. 4 Ch1-p10
Table 1.1 Basic Statement Forms 5 Ch1-p10

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Table 1.2 Truth Table Defining 6 Ch1-p11
Table 1.3 Truth Table Defining and 7 Ch1-p11

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Conditional Statement If e>3, then e 2 >9. p= e>3 q= e 2 >9 p q T T T (e=7) F F T (e=2) If e>3, then e 2 >36. p= e>6 q= e 2 >9 p q T T T (e=7) F T T (e=5)
Precedence of Operations The order of precedence of the basic operations listed from highest to lowest is 9 Ch1-p11

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EXAMPLE 1.3 Make a truth table for the statement form p q r. Solution. 10 Ch1-p12
EXAMPLE 1.4 Make a truth table for the statement form ¬( p q ). Solution. 11 Ch1-p12

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DEFINITION 1.2 (a) The exclusive or operation is defined by p q = ( p q ) ¬( p q ). (b) The if and only if operation ↔ is defined by p q = ( p →q) ( q p ). Note that iff is also used to denote ↔. 12 Ch1-p12
Table 1.4 Truth Table Defining and↔ 13 Ch1-p13

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DEFINITION 1.3 (a) A tautology is a statement form that is always true. We denote a tautology by t . (b) A contradiction is a statement form that is always false. We denote a contradiction by f . 14 Ch1-p13
EXAMPLE 1.5 (a) p ¬ p is a tautology. Solution. In the truth table all of the entries in the column for p p are T . 15 Ch1-p13

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EXAMPLE 1.5 (b) p ¬ p is a contradiction. Solution. In the truth table all of the entries in the column for p p are F. 16 Ch1-p13
DEFINITION 1.4 Two statement forms p and q are logically equivalent , written p q , if and only if the statement form p q is a tautology. 17 Ch1-p14

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Verify that ¬( p q ) ≡ p ¬ q . Solution. Since (¬( p q )) ↔ ( p ¬ q ) is a tautology, we conclude that ¬( p q ) ≡ p ¬ q . It would also suffice to confirm that ¬(
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## This note was uploaded on 11/11/2011 for the course MATH 110 taught by Professor Staff during the Winter '08 term at BYU.

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CH1 (2010)_0304 - Discrete Mathematics 1st Edition Kevin...

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