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Section 2.2

Section 2.2. Indirect Proof: Uses Laws of Logic to Prove Conditional Statements True or False. 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

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Section 2.2

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  1. Section 2.2 Indirect Proof: Uses Laws of Logic to Prove Conditional Statements True or False Section 2.2

  2. 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.” Section 2.2

  3. Conditional and its Inverse and Converse In general, the inverse and converse of a given conditional need notbe 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. Section 2.2

  4. Conditional and its ContrapositiveThe 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 Section 2.2

  5. 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: • Suppose that ~Q is true. • Reason from the supposition until you reach a contradiction. • Note that the supposition claiming that ~Q is true must be false and that Q therefore must be true. Section 2.2

  6. 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 thatr|| s . Ex. 5 p. 84 Example of Indirect Proof / / / / Section 2.2

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