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1 Section 1.2 Propositional Equivalences

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2 Equivalent Propositions Have the same truth table Can be used interchangeably For example, exclusive or and the negation of biconditional are equivalent propositions: pq p qp q (p q) TT F T F TF T F T FT T F T FF F T F

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3 Equivalent propositions Logical equivalence is denoted with the symbol If p q is true, then p q

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4 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 pp p p T F T F T T

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5 Contradiction A compound proposition that is always false For example, p p is a contradiction: p pp p p T F F F T F

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6 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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7 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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8 Method 1 example Show that p q p q pq p q q p qq TT F F F F TF F T T T FT T T F T FF T F T F

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9 Method 1 example Show that (p q) p q pqp q (p q) pp qq p qq TT T F F F F TF F T F T T FT F T T F T FF F T T T T

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10 Method 1 example Show that p (q r) (p q) (p r) pqr qrqrp (q r)pqpqprpr(p q) (p r) TTT T T T T T TTF T T T F T TFT T T F T T TFF F F F F F FTT T F F F F FTF T F F F F FFT T F F F F FFF F F F F F

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11 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 2 N rows

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12 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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13 Identity Laws p T p p F p In other words, if p is ANDed with another proposition known to be true, or ORed with another proposition known to be false, the truth value of the compound proposition will be the truth value of p

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14 Domination Laws p T T p 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 it is composed of any proposition p ANDed with a proposition known to be false

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15 Idempotent Laws p p p p p p A compound proposition composed of any proposition p combined with itself via conjunction or disjunction will have the truth value of p

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16 Double negation ( p) p The negation of a negation is … well, not a negation

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17 Commutative Laws p q q p p q q p Ordering doesn’t matter in conjunction and disjunction (just like addition and multiplication)

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18 Associative Laws (p q) r p (q r) (p q) r p (q r) Grouping doesn’t affect outcome when the same operation is involved - this is true for compound propositions composed of 3, 4, 1000 or N propositions

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19 Distributive Laws p (q r) (p q) (p r) p (q r) (p q) (p r) OR distributes across AND; AND distributes across OR

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20 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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21 Two Laws with No Name p p T p p F A proposition ORed with its negation is always true; a proposition ANDed with its negation is always false

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22 A Very Useful (but nameless) Law (p q) ( p q) The implication “if p, then q” is logically equivalent to NOT p ORed with q

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23 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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24 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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25 Example 2: Show that p 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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26 Section 1.2 Propositional Equivalences - ends -

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