Discrrete mathematics for computer science propositional logic

Propositional Logic• A proposition is a statement that is either true or false.. • We give propositions names such as p, q, and r... • Building propositions from propositions• “And” • S

Propositional Logic

Propositional Logic

• A proposition is a statement that is either true

or false

• We give propositions names such as p, q, and r.

• E.g p = “pigs have wings”

• q = “Obama is U.S president”

• In this world anyway, p is false and q is true.

Truth Tables

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• Building propositions from propositions

• “And”

• So “pigs have wings and Obama is US president” …

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Logical Operators/Connectives

• “Or”

• So “pigs have wings or Obama is US president” …

• “Or” is ambiguous in English but not in Propositional Logic

• E.g., “would you like coffee or ice cream?”

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Logical Operators/Connectives

 “Exclusive or”

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Logical Operators/Connectives

Some expressions can be defined in terms of others

 p ⊕ q is equivalent to (p q) ¬(p q)∨ ∧ ∧

 What does this actually mean?

 That the truth value of the formula on the left is the same as the truth value of the formula on the

right regardless of the truth values of p and q

Propositional Logic

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¬(p q) ∧

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p q or p→q or p q⇒ ⊃

is equivalent to

¬p q∨

So “If pigs have wings then Obama is US president” is true in this world

In fact a false proposition implies any proposition

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“Iff” or logical equivalence

• p ⇔ q means that p and q have the same truth value

• Equivalent to (p ⇒ q) (∧ q ⇒ p)

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