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Section 2.2
Indirect Proof:
Uses Laws of Logic to Prove
Conditional Statements True or False
Section 2.2
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Forms of Indirect Proof
• Conditional (or Implication)
– PQ
– “If it is a wheel, then it is round.”
• Converse of Conditional
– Q P
– “if it is round, then it is a wheel.”
• Inverse of Conditional
– ~P  ~Q
– “If it is not a wheel, then it is not round.”
• Contrapositive of Conditional
– ~Q  ~P
– “If it is not round, then it is not a wheel.”
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Conditional and its Inverse and Converse
In general, the inverse and converse of a given
conditional need not be true when the
conditional is true.
– Conditional:
• If Tom lives in San Diego, then Tom lives in
California.
– Inverse:
• If Tom does not live in San Diego, then Tom does
not live in California.
– Converse:
• If Tom lives in California, the Tom lives in San
Diego.
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Conditional and its Contrapositive
The Law of Negative Inference
• The contrapositive of a given conditional is
always true when the conditional is true.
• A conditional statement can always be
replaced with its contrapositive.
– Conditional:
• If two angles are supplementary, then the sum of
the angles is 180.
– Contrapositive:
• If the sum of two angles is not 180, then the two
angles are not supplementary
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Indirect Proof
Law of Negative Inference (Contraposition)
•
Although direct proofs (2-column) are the most common
type of proofs, some theorems are more easily proved
using the format of an indirect proof.
p. 82.
P →Q
If Erin gets paid, she will go to the concert
~Q
Erin didn’t go to the concert
∴ ~P
Erin didn’t get paid.
Strategy:
1. Suppose that ~Q is true.
2. Reason from the supposition until you reach a contradiction.
3. Note that the supposition claiming that ~Q is true must be
false and that Q therefore must be true.
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Example of Indirect Proof
Prove: If two lines are cut by a
transversal so that corresponding
angles are not congruent, then the
two lines are not parallel.
Given:
r and s are cut by transversal t.
1 / 5
Prove: r ||/ s
Assume that r || s. When they are
cut
by the transversal, corresponding
/
angles are congruent. But 1 ≢ 5
by hypothesis. Thus the assumed
statement that r || s is false. It follows
that r ||/ s .
Ex. 5 p. 84
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