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Math31LectureNotes

# Math31LectureNotes - Math 31 Lecture Notes test author Fall...

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Math 31 Lecture Notes test author Fall 2006

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Contents 1 Propositional Logic 2 1.1 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Excercises . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 The Conditional and Biconditional Connectives . . . . . . . . 6 2 Predicate Logic 8 3 Methods of Proof 9 4 Mathematical Induction 10 1
Chapter 1 Propositional Logic We will be concerned here with the essential form of valid and invalid argue- ments. 1.1 Propositions A proposition is a sentence or formula 1 which is true or false, but not bot. Exercises Consider the following sentances and formals 1. 10 < 7 2. Why is mathematical induction important? 3. x + y = 5 4. 10 10000 can be written as the sum of two primes. 5. 2 + 2 = 4 Only a, d, and e are propositions, though we may not know the truth value of f(d). Sentence b is a question, and has no truth value. Formula c is neither true nor false until the variables x and y are specified. We typically symbolize propositions by p, q, r, ... or P, Q, R, ..., or ... 1 Actually, a well-formed formula, or wff for short 2

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CHAPTER 1. PROPOSITIONAL LOGIC 3 1.2 Connectives We can build more complicated compound propositions from simples propo- sitions p,q,... . We use logical connectives to do this. The three simples connectives are ’not’, ’or’, and ’and’, which are symbolized as shown below. Symbol Name Meaning 2 Negation Not Disjunction Or Conjunction And If p and q are propositions, we can form the compound propositions p , p q , and p q . We call p and q the atomic propositions (or atoms ) of the compound propositions. The truth value of p , p q , and p q in terms of the truth values of the atoms p and q are given in the following truth tables . p p T F F T p q p q T T T T F T F T T F F F p q p q T T T T F F F T F F F F Other common connectives: Symbol Meaning Exclusive Or (XOR) | Not And (NAND) Not Or (NOR) The truth table for these connectives is given by p q p q p | q p q T T F F F T F T T F F T T T F F F F T T
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