Logic and Truth Tables

1 Math 101 – Final Exam “Walk-through” Review of Logic

Logic Symbol Summary Operator NOT AND OR IMPLIES Symbols ~ p p  q p  q pq Descriptio Negatio Conjunction Disjunction Conditional n n

Truth Table p ~p p q p  q p q p  q p q p → q T F T T T T T T T T T F T T F F T F T T F F F T F F T T F T T F F F F F F F F T

Venn Diagram

Equivalent Statements: Their truth tables have the same ______means “is equivalent to” DeMorgan’s Law: a) ~ (p  q) ~ p  ~ q (negation of AND) b) ~ (p  q) ~ p  ~ q (negation of OR) Ex a Use truth tables to demonstrate ~ (p  q) is equivalent to ~ p  ~ q p q

Negations of the other 2 operators a) ~ (~p) (negation of NOT) b) ~ (pq) (negation of IMPLIES) Ex b Use truth tables to test whether ~ (pq) is equivalent to p  ~q p q

Conditionals: “If” vs. “Only If”

If p then q (Intuitive order) 2

The word “If” (alone) points to the ______immediately after it.

Ex: You are not a vegetarian if you eat meat.

Only if q then p p only if q (Less intuitive order)

The phrase “Only if” points to the ______immediately after it.

Ex: You can graduate only if you pass math.

Ex Give the premise and conclusion. Then arrange statements into “if p, then q” format.

1) You are not a vegetarian if you eat meat.

2) I ride my bike only if it’s not raining.

3) Only if you are committed will you stay married.

Biconditional: “If and Only If” Symbolically: (pq)  (qp) Shortcut: p  q 3 Variations on Conditionals and Equivalance

Conditional Statement: pq Contrapositive: ~ q~ p

Inverse: ~ p  ~q Converse: qp

Ex “I ride my bike only if it’s not raining. ” 1) Write the contrapositive, inverse & converse

2) Write the negation of the statement (hint: see page 1, negations of other operators)

Necessary and Sufficient Conditions Cheap Trick:  Sufficient:

Ex Translate to symbols: Being a chimp is sufficient for not being a monkey.

Venn Diagram:

 Necessary:

Ex Translate to symbols: Not being a monkey is necessary for being an ape

Venn Diagram:

Ex For each of the statements 1) Assign “p” and “q” to phrases, 2) Translate to symbols, 3) Rewrite the statement in “If p, then q” format, and 4) show the Venn Diagram a) Warm weather is sufficient for no snow. b) Clouds are necessary for snow. 4 Valid Arguments

Some, All, and None Using Venn Diagrams

Ex Determine the validity using a Venn Diagram. 1. No vegetarian owns a gun. 2. All policemen own guns Therefore, no policeman is a vegetarian

Bonus: Based on your Venn Diagram, write an “If p then q” conditional statement

Bonus2: Based on your Venn Diagram, write statements using “necessary” & “sufficient”

Ex Determine the validity using a Venn Diagram 1. All poets are eloquent. 2. Some poets are wine connoisseurs Therefore some wine connoisseurs are eloquent

Determining Validity with Truth Tables

Hypothesis 1: If I’m not old enough, I can’t drink Hypothesis 2: I’m old enough Conclusion: Therefore, I can drink

Notation:

Use a truth table to determine whether the argument: above is valid.

p q 1. T T 2. T F 3. F T 4. F F