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A sample class of Discrete Structures - PowerPoint PPT Presentation

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

(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.

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.”);}

• Let p = (dScore <= 100),

• q = (dScore > 90) .

• We write p q

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

• Let p = (dScore <= 100);

• q = (dScore > 90).

• The expression can be translated to p q

• p=true;

• q=true.

• By truth table,

• 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.

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

• We write: p→q;

• p=true;

• q=false.

• By truth table,

• 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.

• Two definitions:

• logical conjunction p ∧ q

• Logical implication p → q

• Truth table of p ∧ q, p → q

• Any questions?