CH1 (2010)_0304

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

Info iconThis preview shows pages 1–19. Sign up to view the full content.

View Full Document Right Arrow Icon
Discrete Mathematics, 1st Edition Kevin Ferland Chapter 1 Logic and Sets 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
Table 1.1 Basic Statement Forms 5 Ch1-p10
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Table 1.2 Truth Table Defining 6 Ch1-p11
Background image of page 6
Table 1.3 Truth Table Defining and 7 Ch1-p11
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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)
Background image of page 8
Precedence of Operations The order of precedence of the basic operations listed from highest to lowest is 9 Ch1-p11
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
EXAMPLE 1.3 Make a truth table for the statement form p q r. Solution. 10 Ch1-p12
Background image of page 10
EXAMPLE 1.4 Make a truth table for the statement form ¬( p q ). Solution. 11 Ch1-p12
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 12
Table 1.4 Truth Table Defining and↔ 13 Ch1-p13
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 14
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
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 16
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
Background image of page 17

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 ¬(
Background image of page 18
Image of page 19
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 63

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

This preview shows document pages 1 - 19. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online