Proofs Using Logical Equivalences

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Proofs Using Logical Equivalences

• Rosen 1.2

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List of Logical Equivalences
p?T ? p p?F ? p Identity Laws p?T ? T
p?F ? F Domination Laws p?p ? p p?p ? p
Idempotent Laws ?(?p) ? p Double Negation
Law p?q ? q?p p?q ? q?p Commutative
Laws (p?q)? r ? p? (q?r) (p?q) ? r ? p ? (q?r)

Associative Laws
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List of Equivalences
p?(q?r) ? (p?q)?(p?r) Distribution Laws p?(q?r)
? (p?q)?(p?r) ?(p?q)?(?p ? ?q) De Morgans
Laws ?(p?q)?(?p ? ?q) Miscellaneous p ? ?p ?
T Or Tautology p ? ?p ? F And
Contradiction (p?q) ? (?p ? q) Implication
Equivalence p?q?(p?q) ? (q?p) Biconditional
Equivalence
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The Proof Process
Assumptions
-Definitions -Already-proved equivalences -Stateme
nts (e.g., arithmetic or algebraic)
Logical Steps
Conclusion (That which was to be proved)
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Prove (p??q) ? q ? p?q

• (p??q) ? q Left-Hand Statement
• ? q ? (p??q) Commutative
• ? (q?p) ? (q ??q) Distributive
• ? (q?p) ? T Or Tautology
• ? q?p Identity
• ? p?q Commutative

Begin with exactly the left-hand side
statement End with exactly what is on the
right Justify EVERY step with a logical
equivalence
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Prove (p??q) ? q ? p?q

• (p??q) ? q Left-Hand Statement
• ? q ? (p??q) Commutative
• (q?p) ? (q ??q) Distributive
• Why did we need this step?

Our logical equivalence specified that ? is
distributive on the right. This does not
guarantee the equivalence works on the left!
Ex. Matrix multiplication is not always
commutative (Note that whether or not ? is
distributive on the left is not the point here.)
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Prove p ? q ? ?q ? ?p
Contrapositive

• p ? q
• ? ?p ? q Implication Equivalence
• ? q ? ?p Commutative
• ? ?(?q) ? ?p Double Negation
• ? ?q ? ?p Implication Equivalence

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Prove p ? p ? q is a tautologyMust show that
the statement is true for any value of p,q.

• p ? p ? q
• ? ?p ? (p ? q) Implication Equivalence
• ? (?p ? p) ? q Associative
• ? (p ? ?p) ? q Commutative
• ? T ? q Or Tautology
• ? q ? T Commutative
• T Domination
• This tautology is called the addition rule of
  inference.

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Why do I have to justify everything?

• Note that your operation must have the same order
  of operands as the rule you quote unless you have
  already proven (and cite the proof) that order is
  not important.
• 34 43
• 3/4 ? 4/3
• AB ? BA for everything (for example, matrix
  multiplication)

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Prove (p?q) ? p is a tautology

• (p?q) ? p
• ? ?(p?q) ? p Implication Equivalence
• ? (?p??q) ? p DeMorgans
• ? (?q??p) ? p Commutative
• ? ?q? (?p ? p) Associative
• ? ?q? (p ? ?p) Commutative
• ? ?q? T Or Tautology
• ? T Domination

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Prove or Disprove

• p ? q ? p ? ?q ???
• To prove that something is not true it is enough
  to provide one counter-example. (Something that
  is true must be true in every case.)
• p q p?q p??q
• F T T F
• The statements are not logically equivalent

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Prove?p ? q ?p ? ?q

• ?p ? q
• ? (?p?q) ? (q??p) Biconditional Equivalence
• ? (??p?q) ? (?q??p) Implication Equivalence (x2)
• ? (p?q) ? (?q??p) Double Negation
• ? (q?p) ? (?p??q) Commutative
• ? (??q?p) ? (?p??q) Double Negation
• ? (?q?p) ? (p??q) Implication Equivalence (x2)
• ? p ? ?q Biconditional Equivalence