§1.2 Propositional Equivalence

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§1.2 Propositional Equivalence - PowerPoint PPT Presentation

§1.2 Propositional Equivalence. Two syntactically ( i.e., textually) different compound propositions may be the semantically identical ( i.e., have the same meaning). We call them equivalent . Learn: Various equivalence rules or laws .

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§1.2 Propositional Equivalence
• Two syntactically (i.e., textually) different compound propositions may be the semantically identical (i.e., have the same meaning). We call them equivalent. Learn:
• Various equivalence rules or laws.
• How to prove equivalences using symbolic derivations.

A tautology is a compound proposition that is trueno matter what the truth values of its atomic (or component) propositions are!

Ex.p  p[What is its truth table?]

A contradictionis a compound proposition that is false no matter what! Ex.p  p[Truth table?]

Other compound props. are contingencies.

Logical Equivalence

p  q

• Compound propositions p and q are logically equivalent to each other (written p q, or pq) IFFp and q contain the same truth values as each other in all rows of their truth tables.
• Question: How many different propositions can be constructed from n propositional variables?
Proving Equivalence via Truth Tables

Ex. Prove that pq  (p  q).

F

T

T

T

F

T

T

F

F

T

T

F

T

F

T

T

F

F

F

T

Proving Equivalence via Abbreviated Truth Tables

Try to find a counter example

Ex. Prove that pq  (p  q).

Case 1: Try left side false, right side true

• Assume p=F andq=F, then (p  q) =F.

Case 2: Try right side false, left side true

• Assume(p  q) F, then (p  q) T, then p  qT, thenp=F andq=F, then pq=F.

Logical Non-Equivalence

Ex. p q and q p are not logically equivalent Prove that.

Equivalence Laws
• Equivalence Laws provide a pattern or template that can be used to match all or part of a much more complicated proposition and to find an equivalence for it.
• Equivalent expressions can always be substituted for each other in a more complex expression - useful for simplification.
Equivalence Laws - Examples
• Identity: pT  p pF  p
• Domination: pT  T pF  F
• Idempotent: pp  p pp  p
• Double negation: p  p
• Commutative: pq  qp pq  qp
• Associative: (pq)r  p(qr) (pq)r  p(qr)
More Equivalence Laws
• Distributive: p(qr)(pq)(pr)p(qr)(pq)(pr)
• De Morgan’s:(p1p2…pn)  (p1p2…pn)(p1p2…pn)  (p1p2…pn)
• Trivial tautology/contradiction:p  p  T p  p  F
• Implication: pq  p q
More Equivalence Laws
• Absurdity: (p q)  (p  q ) p
• Contrapositive: pq  q  p
• Absorption: p  (p  q)  p p  (p  q)  p
• Exportation: (p  q)  r  p (q  R)
Defining Operators via Equivalences

Using equivalences, we can define operators in terms of other operators:

• Exclusive or: p q  (pq)  (pq)p q  (pq)  (qp)
• Implies: pq p q
• Biconditional: pq (pq) (qp)pq (p q)

p  q  (p  q)  ( p  q)

Tautology Example

Demonstrate that

[¬p(p q )]q

is a tautology in two ways:

• Using a truth table (did above)
• Using a proof relying on “Equivalence Laws” to derive True through a series of logical equivalences
Tautology by proof

[¬p(p q )]q

 [(¬pp)(¬pq)]q Distributive

 [ F  (¬pq)]q Trivial Contradiction

 [¬pq ]q Identity

¬ [¬pq ] q Implies

 [¬(¬p)¬q ] q DeMorgan

 [p ¬q ] q Double Negation

p  [¬q q ]Associative

p  [q ¬q ]Commutative

p  T Trivial Tautology

 T Domination

Normal or Canonical Forms
• Normal or Canonical Forms：Unique representations of a proposition
• Examples: Construct a simple proposition of two variables which is true only when
• P is true and Q is false:
• P is true and Q is true:
• P is true and Q is false or P is true and Q is true:
Disjunctive Normal Form

A disjunction of conjunctions where

• every variable or its negation is represented once in each conjunction (a minterm)
• each minterms appears only once
• Important in switching theory, simplification in the design of circuits.
To find the minterms of the DNF
• Use the rows of the truth table where the proposition is 1 or True
• If a zero appears under a variable, use the negation of the propositional variable in the minterm
• If a one appears, use the propositional variable.
• Example: Find the DNF of
Example
• Find the DNF of
• There are 5 cases where the proposition is true, hence 5 minterms. Rows 1,2,3, 5 and 7 produce the following disjunction of minterms:
Conjunctive Normal Form

Similarly, Conjunctive Normal Form is a conjunction of disjunctions where

• every variable or its negation is represented once in each disjunction (a maxterm)
• each maxterms appears only once
To find the maxterms of the CNF
• Use the rows of the truth table where the proposition is 0 or False
• If a one appears under a variable, use the negation of the propositional variable in the maxterm
• If a zero appears, use the propositional variable.
• Example: Find the CNF of
Example
• Find the CNF of
• There are 3 cases where the proposition is false, hence 3 maxterms. Rows 4, 6 and 8 produce the following conjunction of maxterms:

(P Q )¬R  (P ¬Q  ¬R) (¬P Q  ¬R) (¬P  ¬Q  ¬R)

Blackboard Exercises for 1.2

Worked out on the black-board.

• “I don’t drink and drive” is logically equivalent to “If I drink, then I don’t drive”
Review: Propositional Logic (§§1.1-1.2)
• Atomic propositions: p, q, r, …
• Boolean operators:      
• Compound propositions: s : (p q)  r
• Equivalences:pq  (p  q)
• Proving equivalences using:
• Truth tables.
• Symbolic derivations. p q  r …
• Next: PREDICATE LOGIC