1 2 propositional equivalence
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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 .

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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

 T Domination

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
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