Logical Conjunction and Implication in Discrete Structures

1. A sample class of Discrete Structures (Logical conjunction and logical implication) Gongjun Yan Computer Science Department, Old Dominion University, Norfolk, VA

2. 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

3. 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." • }

4. Today’s Content • Logical Conjunction (AND), logical implication (IF-Then) • The truth table of AND and IF-THEN

5. 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

6. 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

7. 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

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. Summary • Two definitions: • logical conjunction p ∧ q • Logical implication p → q • Truth table of p ∧ q, p → q • Any questions?

14. 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