MAT 251 Discrete Mathematics

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MAT 251 Discrete Mathematics

• Logic and Proofs

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Section 1.2 Propositional Equivalences

• Def A compound proposition that is always
  true, no matter what the truth values of the
  propositions that occur in it, is called a
  tautology. A compound proposition that is always
  false is called a contradiction. A compound
  proposition that is neither a tautology nor a
  contradiction is called a contingency.

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Section 1.2 Propositional Equivalence

• Consider p ? p and p ? p. What can you
  conclude about these propositions?

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Section 1.2 Propositional Logic

• Def Compound propositions that have the same
  truth values in all possible cases are called
  logically equivalent. This can be shown also if p
  ? q is a tautology.
• If p and q are equivalent, then we write that
  p ? q.

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Section 1.2 Propositional Equivalence

• Show that (p ? q) and p ? q are logically
  equivalent.

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Section 1.2 Propositional Equivalence

• Show that p ? q and q ? p are logically
  equivalent.

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Section 1.2 Propositional Logic

• On pages 24 and 25 you can find all the Logical
  Equivalences with special significance and names.
• Using the above equivalences one can construct
  new equivalences as follows

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Section 1.2 Propositional Equivalence

• Page 29/26
• Show that p ?(q ? r) and
• q ?(p ? r) are logically equivalent.
• Solution
• p ? (q ? r) ? p ? (q ? r) (T 7 3)
• ? p ? (q ? r) (T 7 1)
• ? (p ? q) ? r (Associative
  Law)
• ? (q ? p) ? r (Commutative Law)
• ? q ? (p ? r) (Associative
  Law)
• ? (q) ? (p ? r) (T 7 3)
• ? q ? (p ? r) (Double Negation)

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Section 1.2 Propositional Logic

• The notes have been created with the use of
  Discrete mathematics and Its Applications, Sixth
  Edition by K. H. Rosen