Does p logically imply c ? Is the proposition (p → c) a tautology? Is the proposition (¬ p ∨ c) is a tautology? Is the proposition (¬ c → ¬ p) is a tautology? Is the proposition (p ∧ ¬ c) is a contradiction?
Let p = h1 ∧ h2 ∧ … ∧ hn . The following propositions are equivalent:
p ⇒ c (p → c) is a tautology. (¬ p ∨ c) is a tautology. (¬ c → ¬ p) is a tautology. (p ∧ ¬ c) is a contradiction.
MSU/CSE 260 Fall 2009
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Direct Direct Contrapositive Contradiction
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Formal Proofs
Formal Proof
A proof is equivalent to establishing a logical implication chain Given premises (hypotheses) h1 , h2 , … , hn and conclusion c, to give a formal proof that the hypotheses imply the conclusion, entails establishing
To prove: h1 ∧ h2 ∧ … ∧ hn ⇒ c
Produce a series of wffs, p1 , p2 , … pn, c such that each wff pr is:
h1 ∧ h2 ∧ … ∧ hn ⇒ c
MSU/CSE 260 Fall 2009
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Prove the theorem:
pr
It is given that n is an odd integer. Thus n = 2k + 1, for some integer k. Thus n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1 2 Therefore, n is odd.
one of the premises or
a tautology, or
an axiom/law of the domain (e.g., 1+3=4 or x > x+1 )
justified by definition, or
logically equivalent to or implied by one or more propositions pk where 1 ≤ k
