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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.

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

p q

- Compound propositions p and q are logically equivalent to each other (written p q, or pq) 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 Abbreviated Truth Tables

Try to find a counter example

Ex. Prove that pq (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 qT, thenp=F andq=F, then pq=F.

We have exhausted all possibilities and not found a counterexample.

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: pT p pF p
- Domination: pT T pF F
- Idempotent: pp p pp p
- Double negation: p p
- Commutative: pq qp pq qp
- Associative: (pq)r p(qr) (pq)r p(qr)

More Equivalence Laws

- Distributive: p(qr)(pq)(pr)p(qr)(pq)(pr)
- De Morgan’s:(p1p2…pn) (p1p2…pn)(p1p2…pn) (p1p2…pn)
- Trivial tautology/contradiction:p p T p p F
- Implication: pq p q

More Equivalence Laws

- Absurdity: (p q) (p q ) p
- Contrapositive: pq 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 (pq) (pq)p q (pq) (qp)
- Implies: pq p q
- Biconditional: pq (pq) (qp)pq (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

[(¬pp)(¬pq)]q Distributive

[ F (¬pq)]q Trivial Contradiction

[¬pq ]q Identity

¬ [¬pq ] 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:pq (p q)
- Proving equivalences using:
- Truth tables.
- Symbolic derivations. p q r …
- Next: PREDICATE LOGIC

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