Chapter 1: The Foundations: Logic and Proofs 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers 1.5 Rules of Inference 1.6 Introduction to Proofs 1.7 Proof Methods and Strategy

Introduction: Nested Quantifiers Nested quantifiers: Two quantifiers are nested if one is within the scope of the other. Example:  x  y(x+y=0)

Introduction: Nested Quantifiers Example: Domain: real number. Addition inverse:  x  y(x+y=0) Commutative law for addition:  x  y(x+y=y+x) Associative law for addition:  x  y  z (x+(y+z)=(x+y)+z)

The Order of Quantifiers Example: Let Q(x,y) denote “x+y=0.” What are the truth values of the quantifications  y  xQ(x,y) and  x  yQ(x,y), where the domain for all variables consists of all real numbers?

Translating Mathematical Statements into Statements Involving Nested Quantifiers Example 7: Translate the statement “Every real umber except zero has a multiplicative inverse.” (A multiplicative inverse of a real number x is a real number y such that xy=1) HW: Example 8, p54

Translating from Nested Quantifiers into English Example 9: Translate the statement  x(C(x)  y(C(y)  F(x,y))) into English, where C(x) is “x has a computer,” F(x,y) is “x and y are friends,” and the domain for both x and y consists of all students in your school. HW: Example 10, p55.

Negating Nested Quantifiers Example 14: Express the statement  x  y(xy=1) negation of the statement so that no negation precedes a quantifier. HW: Example 15, p57