1. 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

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

3. 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)

4. 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?

6. 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

7. 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.

8. 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