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A sample class of A sample class of Discrete Discrete

StructuresStructures

Gongjun YanGongjun Yan

Computer Science Department,Computer Science Department,

Old Dominion University, Old Dominion University, Norfolk, VANorfolk, VA

(Logical conjunction (Logical conjunction and logical and logical implication)implication)

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ReviewReview

PropositionProposition Declarative statement (no question, no imperative Declarative statement (no question, no imperative

statement)statement) Either true or false but not bothEither true or false but not both

ExamplesExamples Which of the following are propositions?Which of the following are propositions?

Today is Friday. Today is Friday. Everyone is happy.Everyone is happy. What time is it? What time is it?

Game Time!Game Time!

True!True!False!

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More QuestionsMore Questions What are the values of propositions below?What are the values of propositions below?

If a fly has no wings, it is called a walk. If a fly has no wings, it is called a walk. If olive oil comes from olives, then baby oil comes If olive oil comes from olives, then baby oil comes

from babies.from babies.

How does the if statement work in the java How does the if statement work in the java code below?code below? dScore = 93;dScore = 93; if ( if ( (dScore <= 100) && (dScore > 90)(dScore <= 100) && (dScore > 90) ) ) {{ print "You got an A."print "You got an A." }}

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Today’s ContentToday’s Content

Logical conjunction (AND)Logical conjunction (AND) Logical implication (IF-Then)Logical implication (IF-Then) The truth table of AND and IF-THENThe truth table of AND and IF-THEN

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Logical conjunction: ANDLogical conjunction: AND

Is an operation on two propositionsIs an operation on two propositions Produces a value of Produces a value of truetrue if and only if if and only if

both of its propositions are trueboth of its propositions are true Keywords: &&, and, &. Keywords: &&, and, &. We often use We often use p qp q

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Truth TableTruth Table

Indicate the value of the logical Indicate the value of the logical expressions.expressions.

The Truth table of The Truth table of p ∧ qp ∧ q if both p and q are true, then p ∧ q is true

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Example 1Example 1 Translate the underlined expression Translate the underlined expression

into logical expression.into logical expression. dScore = 93;dScore = 93;

if ( if ( (dScore <= 100) && (dScore > 90) (dScore <= 100) && (dScore > 90) )){{

print(“you got an A.”);print(“you got an A.”);}} Let p = (dScore <= 100);Let p = (dScore <= 100); q = (dScore > 90).q = (dScore > 90). p qp q

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Logical implication: Logical implication: IMPLIESIMPLIES

Is a proposition “if p, then q”.Is a proposition “if p, then q”. Produces Produces all trueall true values values except when except when

p is true and q is falsep is true and q is false. . Keywords: if-then, if, then, whenever, Keywords: if-then, if, then, whenever,

only if, etc. only if, etc. We often use p qWe often use p q The Truth table of The Truth table of p → qp → q All true

except when p is true and q is false.

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Example 2Example 2 2.1 Translate English into logical 2.1 Translate English into logical

implication p → q:implication p → q: If olive oil comes from olives, then baby oil comes If olive oil comes from olives, then baby oil comes

from babies.from babies.

AnswerAnswer: : Let p = “Olive oil comes from olives.”Let p = “Olive oil comes from olives.” q = “Baby oil comes from babies.”q = “Baby oil comes from babies.” We write: p q.We write: p q.

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Example 2Example 2 2.2 What is the value of the statement:2.2 What is the value of the statement:

If olive oil comes from olives, then baby oil comes If olive oil comes from olives, then baby oil comes from babies.from babies.

Answer:Answer: Let p = “Olive oil comes from olives.”Let p = “Olive oil comes from olives.”

q = “Baby oil comes from babies.” q = “Baby oil comes from babies.” We write: p → q;We write: p → q; p=true;p=true; q=false.q=false. By truth table, By truth table, the answer is false.the answer is false.

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QuizQuiz What are the values of propositions?What are the values of propositions?

1. Tom Cruise is an actor and he is handsome.1. Tom Cruise is an actor and he is handsome.

pp = “ = “Tom Cruise is an actor.”Tom Cruise is an actor.”

q = “Tom Cruise is handsome.”q = “Tom Cruise is handsome.” AnswerAnswer: We write : We write p q. ∧p q. ∧ p=true.p=true. In my wife’s opinion, q=true. In my wife’s opinion, q=true. By truth table, the answer is By truth table, the answer is truetrue..

2. If Tiger Woods is studying discrete structures, then he will not play 2. If Tiger Woods is studying discrete structures, then he will not play golf.golf.p = “Tiger Woods is studying discrete structures.”p = “Tiger Woods is studying discrete structures.”q = “Tiger Woods will not play golf.”q = “Tiger Woods will not play golf.”

AnswerAnswer: We write : We write p → q. p → q. In my mind, p=false;In my mind, p=false; q=false. q=false. By truth table, the answer is By truth table, the answer is truetrue..

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SummarySummary

Two definitionsTwo definitions:: logical conjunction p ∧ qlogical conjunction p ∧ q Logical implication p → qLogical implication p → q

Truth table of p ∧ q, p → qTruth table of p ∧ q, p → q

Any questions?Any questions?

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More InformationMore Information Course home page linkCourse home page link

http://www.cs.odu.edu/~ygongjun/courses/http://www.cs.odu.edu/~ygongjun/courses/cs381fall09/instructions/ cs381fall09/instructions/

HomeworkHomework: reading and playing!: reading and playing! Translation between p ∧ q, p → q and EnglishTranslation between p ∧ q, p → q and English

Link: Link: course-home-page/if_then.htmlcourse-home-page/if_then.html Truth TablesTruth Tables

Link: Link: course-home-page/truth_table.htmlcourse-home-page/truth_table.html Office hours: Monday 10:00-12:00amOffice hours: Monday 10:00-12:00am

Wednesday 10:00-12:00am Wednesday 10:00-12:00am