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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

- 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.

- Which of the following are propositions?

Value = True

Value = False

- 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."
- }

- Logical Conjunction (AND), logical implication (IF-Then)
- The truth table of AND and IF-THEN

- 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

- 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

- 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

- 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.

- 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.

- 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.

- 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.

- 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.

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

- Two definitions:
- logical conjunction p ∧ q
- Logical implication p → q

- Truth table of p ∧ q, p → q
- Any questions?

- 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

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