Sec 3.3 The Conditional & Circuits
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1 Sec 3.3 The Conditional & Circuits Conditional statement: connective if... then. a compound statement that uses the Conditional statements are also known as implications, and can be written as: p q (pronounced p implies q ) The statement p is called the antecedent. The statement q is called the consequent , N. Van Cleave 1
2 Conditional Examples If you are not home by midnight, (then) you ll be grounded. If he hits a home run, (then) he ll beat the old record. If you scratch my back, (then) I ll scratch yours. If you exceed the speed limit, (then) you ll get a ticket. The English are bad cooks. translation: If you are English, then you are a bad cook. College students are immature. translation: If you are a student, then you are immature , N. Van Cleave 2
3 Truth Table for Conditional Statements There are four possible combinations of truth values for the two component statements p q p q T T? T F? F T? F F? Let s consider: If you are not home by midnight, then you ll be grounded. Is the implication true when: 1. You are not home by midnight and you are grounded 2. You are not home by midnight but you are not grounded 3. You are home by midnight but you are grounded 4. You are home by midnight and you are not grounded , N. Van Cleave 3
4 Another Example Let s consider: If he hits a home run, then he ll beat the old record. p q p q T or F? he hits a home run he hits a home run he doesn t hit a home run he doesn t hit a home run he beats the old record he doesn t beat the old record he beats the old record he doesn t beat the old record , N. Van Cleave 4
5 Another Example How about: If you are English, then you are a bad cook. p q p q T or F? you are English you are English you aren t English you aren t English you are a bad cook you are not a bad cook you are a bad cook you are not a bad cook , N. Van Cleave 5
6 Another Example And finally: If you are a college student, then you are immature. p q p q T or F? you are a college student you are a college student you aren t a college student you aren t a college student you are immature you aren t immature you are immature you aren t immature , N. Van Cleave 6
7 Truth Table for the Conditional If p, then q p q p q T T T T F F F T T F F T If the moon is made of green cheese,... If my name isn t < My name here >... If I finish my homework,... If I had a million dollars,... If wishes were fishes, , N. Van Cleave 7
8 Notes p q is false only when the antecedent is true and the consequent is false If the antecedent is false, then p q is automatically true If the consequent is true, then p q is automatically true , N. Van Cleave 8
9 true or false? true (6 = 6) true (6 = 3) false (6 = 6) false (6 = 3) (6 = 6) true (6 = 3) true (6 = 6) false (6 = 3) false Let p, q, and r be false ( p q) ( p q) ( r q ) ( p q) ( r q ) , N. Van Cleave 9
10 Truth Table: ( p q) ( p q) p q p q p q p q ( p q) ( p q) T T T F F T F F Truth Table: (p q) ( p q) p q p q p p q ( p q ) ( p q ) T T T F F T F F Tautology: a statement that is always true, no matter what the truth values of the components , N. Van Cleave 10
11 Truth Table: p p p p p p T F Truth Table: p p p p p p T F , N. Van Cleave 11
12 Truth Table: ( p q) (q p) p q p q p q (q p) ( p q) (q p) T T T F F T F F Truth Table: Negation of p q p q p q (p q) q p q T T T F F T F F Recall: You are not home by midnight, you are not grounded... the only false result, and thus the negation , N. Van Cleave 12
13 The negation of p q is p q Write the negation of each statement If you are not home by midnight, then you ll be grounded. If he hits a home run, (then) he ll beat the old record. If you scratch my back, (then) I ll scratch yours. If you exceed the speed limit, (then) you ll get a ticket , N. Van Cleave 13
14 If it s Smucker s, it s got to be good! If that is an authentic Persian rug, I ll be surprised. The English are bad cooks. translation: If you are English, then you are a bad cook. College students are immature. translation: If you are a student, then you are immature , N. Van Cleave 14
15 p q is equivalent to p q Rewrite as a statement that doesn t use the if... then connective If you are not home by midnight, then you ll be grounded. If he hits a home run, (then) he ll beat the old record. If you scratch my back, (then) I ll scratch yours. If you exceed the speed limit, (then) you ll get a ticket , N. Van Cleave 15
16 If it s Smucker s, it s got to be good! If that is an authentic Persian rug, I ll be surprised. If you give your plants tender, loving care, they flourish. If she doesn t, he will. If you are a student, then you are immature , N. Van Cleave 16
17 CIRCUITS When will current flow through the switch and wire? A Switch On or Off p p A Series Circuit q q A Parallel Circuit , N. Van Cleave 17
18 What is the corresponding logic statement? p q ~q p q p r p q r ~p ~q r , N. Van Cleave 18
19 Equivalent Statements Used to Simplify Circuits p T T p F F p p T p p F p p p p p p (p q) p q (p q) p q p (q r) (p q) (p r) p (q r) (p q) (p r) p q q p p q p q , N. Van Cleave 19
20 Draw a circuit for: p ( q r) Draw a circuit for p (q r). (Rewrite it first) , N. Van Cleave 20
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