1. 12/6/2015 1 A sample class of Discrete Structures Gongjun Yan Computer Science Department, Old Dominion University, Norfolk, VA (Logical conjunction and logical implication)

2. 12/6/2015 2 Review Proposition Proposition Declarative Declarative Either true or false but not both Either true or false but not both Examples Examples Which of the following are propositions? Which of the following are propositions? Today is Friday. Today is Friday. 1 + 1 = 3 1 + 1 = 3 What time is it? What time is it? Close the window. Close the window. Value = True Value = False

3. 12/6/2015 3 More Questions What are the values of propositions below? What are the values of propositions below? If Tiger Woods is studying discrete structures, then he will not play golf. 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 a fly has no wings, it can be called a walk. If olive oil comes from olives, then baby oil comes from babies. If olive oil comes from olives, then baby oil comes from babies. How the if statement works in the below code? How the if statement works in the below code? dScore = 93; dScore = 93; if ( (dScore 90) ) if ( (dScore 90) ) { print "You got an A." print "You got an A." }

4. 12/6/2015 4 Today’s Content Logical Conjunction (AND), logical implication (IF-Then) Logical Conjunction (AND), logical implication (IF-Then) The truth table of AND and IF-THEN The truth table of AND and IF-THEN

5. 12/6/2015 5 Logical conjunction: AND Is an operation on two propositions Is an operation on two propositions Produces a value of true if and only if both of its propositions are true Produces a value of true if and only if both of its propositions are true Is written as p AND q, p ∧ q, p & q, or p.q Is written as p AND q, p ∧ q, p & q, or p.q We often use p q We often use p q

6. 12/6/2015 6 Truth Table Indicate the value of the logical expressions. Indicate the value of the logical expressions. The # of rows = 2 to the power of (# of operators) The # of rows = 2 to the power of (# of operators) The Truth table of p ∧ q The Truth table of p ∧ q if both p and q are true, then p ∧ q is true

7. 12/6/2015 7 Example Translate the underlined expression into the conjunction p ∧ q Translate the underlined expression into the conjunction p ∧ q dScore = 93; if ( (dScore 90) ) { print(“you got an A.”); } dScore = 93; if ( (dScore 90) ) { print(“you got an A.”); } Answer: Answer: Let p = (dScore <= 100), Let p = (dScore <= 100), q = (dScore > 90). q = (dScore > 90). We write p q We write p q

8. 12/6/2015 8 Example 2 dScore = 93; what is the value of the expression (dScore 90) ? dScore = 93; what is the value of the expression (dScore 90) ? Answer: Answer: Let p = (dScore <= 100); Let p = (dScore <= 100); q = (dScore > 90). q = (dScore > 90). The expression can be translated to p q The expression can be translated to p q p=true; p=true; q=true. q=true. By truth table, By truth table, the answer is true. the answer is true.

9. 12/6/2015 9 Logical implication: IMPLIES Is a proposition “if p, then q”. Is a proposition “if p, then q”. Produces all true values except when p is true and q is false. Produces all true values except when p is true and q is false. Is written as p IMPLIES q, p → q, p ⇒ q Is written as p IMPLIES q, p → q, p ⇒ q We often use p q We often use p q The Truth table of p → q The Truth table of p → q All true except when p is true and q is false.

10. 12/6/2015 10 Example 3 3.1 Translate English into logical implication p→q: 3.1 Translate English into logical implication p→q: If olive oil comes from olives, then baby oil comes from babies. If olive oil comes from olives, then baby oil comes from babies. Answer: Let p = “Olive oil comes from olives.” Answer: 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.

11. 12/6/2015 11 Example 3 3.2 What is the value of the statement: 3.2 What is the value of the statement: If olive oil comes from olives, then baby oil comes from babies. If olive oil comes from olives, then baby oil comes from babies. Answer: Answer: Let p = “Olive oil comes from olives.” q = “Baby oil comes from babies.” Let p = “Olive oil comes from olives.” 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.

12. 12/6/2015 12 Quiz 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. p = “Tom Cruise is an actor.” q= “Tom Cruise is handsome.” Answer: We write p ∧ q. Answer: We write p ∧ q. In my wife’s opinion, p=true, q=true. In my wife’s opinion, p=true, q=true. By truth table, the answer is 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.” 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. Answer: We write p → q. In my mind, p=false, q=false. In my mind, p=false, q=false. By truth table, the answer is true. By truth table, the answer is true.

13. 12/6/2015 13 Summary Two definitions: Two definitions: logical conjunction p ∧ q logical conjunction p ∧ q Logical implication p → q Logical implication p → q Truth table of p ∧ q, p → q Truth table of p ∧ q, p → q Any questions? Any questions?

14. 12/6/2015 14 Information Course home page link Course home page link http://www.cs.odu.edu/~ygongjun/courses/cs 381fall09/instructions/ http://www.cs.odu.edu/~ygongjun/courses/cs 381fall09/instructions/ http://www.cs.odu.edu/~ygongjun/courses/cs 381fall09/instructions/ http://www.cs.odu.edu/~ygongjun/courses/cs 381fall09/instructions/ Homework: finish the online exercises: Homework: finish the online exercises: Translation between p ∧ q, p→q and English Link: course-home-page/if_then.html Translation between p ∧ q, p→q and English Link: course-home-page/if_then.htmlcourse-home-page/if_then.html Turth Tables Link: course-home-page/if_then.html Turth Tables Link: course-home-page/if_then.htmlcourse-home-page/if_then.html Office hours: Monday 10:00-12:00am Wednesday 10:00-12:00am Office hours: Monday 10:00-12:00am Wednesday 10:00-12:00am