Logic, Sets - Propositional Equivalences - Lecture Slides | ENGR 213, Study notes of Discrete Structures and Graph Theory

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Logic, Sets^ Section 1.2 Propositional Equivalences

Propositional Equivalences^

Definitions „^ A^ tautology^ is a proposition which is alwaystrue.Classic Example:

P∨¬P

„^ A^ contradiction

is a proposition which is

always false.Classic Example:

: P∧¬P

„^ A^ contingency

is a proposition which neither atautology nor a contradiction.Example: (P ∨Q)→¬R

Propositional Equivalences^

Logical Equivalence: Proof „^ To show logical equivalence, the left side andthe right side must have the same truthvalues independent of the truth value of thecomponent propositions. „^ Example:Show that^ P →

Q^ ⇔ ¬P∨Q

Propositional Equivalences^

Logical Equivalence: Proof „^ To show a proposition is not a tautology ortwo propositions are not logical equivalent:use an^ abbreviated

truth table.

„^ Try to find a^ counter example

or to^ disprove

the assertion. „ Search for a case where the proposition isfalse. „ Example: find out either the two expressionare logical equivalent:

P → Q and P ∨ ¬

Q

Propositional Equivalences^

Logical Equivalence Table Some famous logical equivalences P^ ∧^ T^ ⇔^ P P^ ∨^ F^ ⇔^ P^

Identity P^ ∨^ T^ ⇔T P^ ∧^ F⇔^ F^

Domination P^ ∨^ P⇔^ P P^ ∧^ P⇔P^

Idempotency ¬(¬P)⇔^ P^

Double negation P^ ∨Q⇔Q∨^ P P^ ∧Q⇔Q∧^ P^

Commutativity

Propositional Equivalences^

Logical Equivalence Table(P^ ∨Q)∨^ R⇔^ P∨^ (Q

∨^ R)

(P^ ∧Q)^ ∧^ R⇔^ P^ ∧^

(Q∧^ R)^

Associativity P^ ∧(Q∨^ R)⇔^ (P^ ∧Q

)∨^ (P^ ∧^ R)

P^ ∨(Q∧^ R)⇔^ (P^ ∨Q

)∧^ (P^ ∨^ R)^ Distributivity ¬(P^ ∧Q)⇔ ¬P^ ∨¬

Q

¬(P^ ∨Q)⇔ ¬P^ ∧¬

Q^ DeMorgan’s laws P→Q⇔¬P^ ∨Q^

Implication P^ ∨¬P⇔^ T^

Tautology

Propositional Equivalences^

Logical Equivalence „^ Equivalent expressions can always besubstituted for each other in a more complexexpression - useful for simplification and logicproof. „^ Example:Prove using known logical equivalences:^ (P→Q)⇔(¬Q

→¬P)