A sample class of discrete structures
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A sample class of Discrete Structures. (Logical conjunction and logical implication). Gongjun Yan Computer Science Department, Old Dominion University, Norfolk, VA. Review. Proposition Declarative Either true or false but not both Examples Which of the following are propositions?

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A sample class of Discrete Structures

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A sample class of discrete structures

A sample class of Discrete Structures

(Logical conjunction and logical implication)

Gongjun Yan

Computer Science Department,

Old Dominion University, Norfolk, VA


Review

Review

  • Proposition

    • Declarative

    • Either true or false but not both

  • Examples

    • Which of the following are propositions?

      • Today is Friday.

      • 1 + 1 = 3

      • What time is it?

      • Close the window.

Value = True

Value = False


More questions

More Questions

  • What are the values of propositions below?

    • If Tiger Woods is studying discrete structures, then he will not play golf.

    • If a fly has no wings, it can be called a walk.

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

  • How the if statement works in the below code?

    • dScore = 93;

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

    • {

    • print "You got an A."

    • }


Today s content

Today’s Content

  • Logical Conjunction (AND), logical implication (IF-Then)

  • The truth table of AND and IF-THEN


Logical conjunction and

Logical conjunction: AND

  • Is an operation on two propositions

  • Produces a value of trueif and only if both of its propositions are true

  • Is written as p AND q, p ∧ q, p & q, or p.q

  • We often use p q


Truth table

Truth Table

  • Indicate the value of the logical expressions.

  • The # of rows = 2 to the power of (# of operators)

  • The Truth table of p ∧ q

if both p and q are true, then p ∧ q is true


Example

Example

  • Translate the underlined expression into the conjunction p ∧ q

  • dScore = 93;if ( (dScore <= 100) && (dScore > 90) ){print(“you got an A.”);}

  • Answer:

    • Let p = (dScore <= 100),

    • q = (dScore > 90) .

    • We write p q


Example 2

Example 2

  • dScore = 93;what is the value of the expression(dScore <= 100) && (dScore > 90) ?

  • Answer:

    • Let p = (dScore <= 100);

    • q = (dScore > 90).

    • The expression can be translated to p q

    • p=true;

    • q=true.

    • By truth table,

    • the answer is true.


Logical implication implies

Logical implication: IMPLIES

  • Is a proposition “if p, then q”.

  • Produces all true values except when p is true and q is false.

  • Is written as p IMPLIES q , p → q, p ⇒ q

  • We often use p q

  • The Truth table of p → q

All true except when p is true and q is false.


Example 3

Example 3

  • 3.1 Translate English into logical implication p→q:

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

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

    • q = “Baby oil comes from babies.”

    • We write: p q.


Example 31

Example 3

  • 3.2 What is the value of the statement:

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

  • Answer:

    • Let p = “Olive oil comes from olives.” q = “Baby oil comes from babies.”

    • We write: p→q;

    • p=true;

    • q=false.

    • By truth table,

    • the answer is false.


A sample class of discrete structures

Quiz

  • What are the values of propositions?

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

      p = “Tom Cruise is an actor.”

      q= “Tom Cruise is handsome.”

    • Answer: We write p ∧ q.

    • In my wife’s opinion, p=true, q=true.

    • By truth table, the answer is true.

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

    • Answer: We write p → q.

    • In my mind, p=false, q=false.

    • By truth table, the answer is true.


Summary

Summary

  • Two definitions:

    • logical conjunction p ∧ q

    • Logical implication p → q

  • Truth table of p ∧ q, p → q

  • Any questions?


Information

Information

  • Course home page link

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

  • Homework: finish the online exercises:

    • Translation between p ∧ q, p→q and EnglishLink: course-home-page/if_then.html

    • Turth TablesLink: course-home-page/if_then.html

  • Office hours: Monday 10:00-12:00am Wednesday 10:00-12:00am


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