1 2 propositional equivalence
This presentation is the property of its rightful owner.
Sponsored Links
1 / 22

§1.2 Propositional Equivalence PowerPoint PPT Presentation


  • 86 Views
  • Uploaded on
  • Presentation posted in: General

§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 .

Download Presentation

§1.2 Propositional Equivalence

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


1 2 propositional equivalence

§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.


Tautologies and contradictions

Tautologies and Contradictions

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

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

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

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.

    We have exhausted all possibilities and not found a counterexample.


Logical non equivalence

Logical Non-Equivalence

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


Equivalence laws

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

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

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 laws1

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

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

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

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

 TDomination


Normal or canonical forms

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

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

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

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

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

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


Example1

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

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

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


  • Login