Post on 17-Oct-2019
Propositional Equivalences
Niloufar Shafiei
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Review New propositions, called compound propositions,
can be formed from existing propositions using logical operations.
Logical operators Negation: ¬p “not p.” Conjunction: p∧q “p and q.” Disjunction: p∨q “p or q.” Exclusive or: p⊕q “p or q, but not both.” Conditional statement:
p→q “If p, then q.” Biconditional statement:
p↔q “p if and only if q.”
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Review The Truth value of a proposition is true,
denoted by T, if it is a true proposition.p: 1+3=4 True
The Truth value of a proposition is false, denoted by F, if it is a false proposition.
p: It snows today. False
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Compound propositions A compound proposition that is always true is called a
tautology.p ∨ ¬p
p: true ¬p: false p ∨ ¬p: truep: false ¬p: true p ∨ ¬p: true
A compound proposition that is always false is called a contradiction.
p ∧ ¬pp: true ¬p: false p ∧ ¬p: falsep: false ¬p: true p ∧ ¬p: false
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Equivalent propositions (review) When two compound propositions always have
the same truth value, they are called equivalent.
Conditional Statementp→qIf it snows, then I stay at home.
Contrapositive of p→q¬q→¬pIf I do not stay at home, then it does not snow.
p q p→qT T TT F FF T TF F T
¬q→¬pTFTT
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Logical equivalence Compound propositions that have the same
truth values in all cases are called logically equivalent.
The compound propositions v and w are called logically equivalent if v↔w is a tautology, denoted by v≡ w.(v↔w is true if v and w have the same truth value.)
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Logical equivalence v↔w is a tautology.
The truth value of v↔w is always true.
v and w always have the same truth value.
v and w are logically equivalent (v≡ w).
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Logical equivalence (example)p∧T ≡ p
p p∧TT TF F
p↔p∧TTT
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Logical equivalence (example)p→q ≡ ¬p∨q
p ¬p∨qT TF TTF
qTTFF
¬pFTFT
FT
p→qTTFT
(¬p∨q)↔(p→q)TTTT
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Truth table of compound proposition For each additional propositional variable,
we need to double the number of rows in the truth tables.
In general, the truth table of a compound proposition that involves n propositional variables has 2n rows.
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Distributive lawsp∨(q∧r) ≡ (p∨q)∧(p∨r)
p q r p∨(q∧r) (p∨q)∧(p∨r) p∨(q∧r) ↔ (p∨q)∧(p∨r)TTT T T TFTT T T TTFT T T TFFT T T TTTF T T TFTF F F TTFF F F TFFF F F T
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Distributive lawsp∨(q∧r)
≡ (p∨q)∧(p∨r) p∧(q∨r)
≡ (p∧q)∨(p∧r)
Example:s∧(t∨r) is logically equivalent to:a. (s∨t)∧(s∨r) b. s∨(t∧r) c. (s∧t)∨(s∧r)
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De Morgan’s laws¬(p∨q)
≡ ¬p ∧ ¬q ¬(p∧q)
≡ ¬p ∨ ¬q
Example:¬s ∧ ¬r is logically equivalent to:a. ¬(s∨r) b. s∨r c. ¬(s∧r)
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De Morgan’s laws
¬(p1 ∨ p2 … ∨ pn) ≡ ¬p1 ∧ ¬p2 … ∧ ¬pn
¬(p1 ∧ p2 … ∧ pn) ≡ ¬p1 ∨ ¬p2 … ∨ ¬pn
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ExampleShow that ¬q∧(¬p∨q) ≡ ¬(p∨q) by developing a series
of logical equivalences.Solution:¬q ∧ (¬p∨q) ≡(¬q∧¬p)∨(¬q∧q) ≡(¬q∧¬p) ∨F ≡(¬q∧¬p) ≡¬(q∨p) ≡¬(p∨q)
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ExampleIs p∨(¬(p∧q)) a tautology?Solution:p∨(¬(p∧q)) ≡p∨(¬p∨¬q) ≡(p∨¬p)∨¬q ≡T∨¬q ≡T
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ExampleUse De Morgan laws to find the negation of the following statement.
Jen walks or takes the bus to the class.Solution: Determine individual propositions
p: Jen walks. q: Jen takes the bus to the class.
Translate the statement to compound proposition p∨q
Find the negation of proposition (using De Morgan law) ¬ (p∨q) ≡
¬p∧¬q Translate the negation of the proposition to English sentence
¬p:Jen does not walk.
¬q: Jen does not take the bus to the class.
¬p∧¬q:Jen does not walk and Jen does not take the bus to the class.
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Recommended exercises1,4,6,7,9,13,17,32, Example 6,7,8