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Section 1.2

Propositional Equivalences

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Equivalent Propositions

• Have the same truth table

• Can be used interchangeably

• For example, exclusive or and the negation of biconditional are equivalent propositions:p q p q p q (p q)

T T F T FT F T F TF T T F TF F F T F

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Equivalent propositions

• Logical equivalence is denoted with the symbol

• If p q is true, then p q

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Tautology

• A compound proposition that is always true, regardless of the truth values that appear in it

• For example, p p is a tautology:

p p p p

T F TF T T

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Contradiction

• A compound proposition that is always false

• For example, p p is a contradiction:

p p p p

T F FF T F

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Tautology vs. Contradiction

• The negation of a tautology is a contradiction, and the negation of a contradiction is a tautology

• Contingency: a compound proposition that is neither a tautology nor a contradiction

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Determining Logical Equivalence

• Method 1: use truth table

• Method 2: use proof by substitution - requires knowledge of logical equivalencies of portions of compound propositions

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Method 1 example

Show that p q p q

p q p p q q p q

T T F F F FT F F T T TF T T T F TF F T F T F

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Method 1 example

Show that (p q) p q

p q p q (p q) p q p q

T T T F F F FT F F T F T TF T F T T F TF F F T T T T

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Method 1 exampleShow that p (q r) (p q) (p r)

p q r qr p(qr) pq pr (pq)(pr)

T T T T T T T TT T F T T T F TT F T T T F T TT F F F F F F FF T T T F F F FF T F T F F F FF F T T F F F FF F F F F F F F

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The limits of truth tables

• The previous slide illustrates how truth tables become cumbersome when several propositions are involved

• For a compound proposition containing N propositions, the truth table would require 2N rows

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Method 2: using equivalences

• There are many proven equivalences that can be used to prove further equivalences

• Some of the most important and useful of these are found in Tables 5, 6 and 7 on page 24 of your text, as well as on the next several slides

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Identity Laws

p T pp F p

In other words, if p is ANDed with another propositionknown to be true, or ORed with another proposition knownto be false, the truth value of the compound propositionwill be the truth value of p

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Domination Laws

p T Tp F F

A compound proposition will always be true if it is composed of any proposition p ORed with any proposition known to be true.

Conversely, a compound proposition will always be false if itis composed of any proposition p ANDed with a proposition known to be false

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Idempotent Laws

p p pp p p

A compound proposition composed of any proposition pcombined with itself via conjunction or disjunction willhave the truth value of p

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Double negation

(p) p

The negation of a negation is … well, not anegation

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Commutative Laws

p q q pp q q p

Ordering doesn’t matter in conjunction and disjunction(just like addition and multiplication)

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Associative Laws

(p q) r p (q r)(p q) r p (q r)

Grouping doesn’t affect outcome when the sameoperation is involved - this is true for compoundpropositions composed of 3, 4, 1000 or N propositions

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Distributive Lawsp (q r) (p q) (p r)p (q r) (p q) (p r)

OR distributes across AND; AND distributesacross OR

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DeMorgan’s Laws

(p q) p q(p q) p q

The NOT of p AND q is NOT p OR NOT q;the NOT of p OR q is NOT p AND NOT q

Like Association, DeMorgan’s Laws apply to N propositions in a compound proposition

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Two Laws with No Name

p p Tp p F

A proposition ORed with its negation is always true;a proposition ANDed with its negation is always false

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A Very Useful (but nameless) Law

(p q) (p q)

The implication “if p, then q” is logicallyequivalent to NOT p ORed with q

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Method 2: Proof by Substitution

• Uses known laws of equivalences to prove new equivalences

• A compound proposition is gradually transformed, through substitution of known equivalences, into a proveable form

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Example 1: Show that(p q) p is a tautology

1. Since (p q) (p q), change compound proposition to: (p q) p

2. Applying DeMorgan’s first law, which states: (p q) p q, change compound proposition to: p q p

3. Applying commutative law: p p q

4. Since p p T, we have T q

5. And finally, by Domination, any proposition ORed with true must be true - so the compound proposition is a tautology

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Example 2: Show thatp q and p q are logically equivalent

1. Start with definition of biconditional:p q p q q p; then the 2 expressions become:(p q) (q p) and (p q) (q p)

2. Since p q p q, change expressions to:((p) q) (q p) and (p q) ((q) p);same as: (p q) (q p) and (p q) (q p)

3. Reordering terms, by commutation, we get:(p q) (p q) and (p q) (p q)

Since the two expressions are now identical, they are clearly equivalent.

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Section 1.2

Propositional Equivalences

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