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Corollary

Purpose: Let’s Define Some Terms!!. CONVERSE. Corollary. Proof by Contradiction . If and only if. Axiom . Implies. Converse of a Statement. Statement: If Rex is a dog then Rex is a mammal. (True). Converse: If Rex is a mammal then Rex is a Dog. (False ).

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Corollary

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  1. Purpose: Let’s Define Some Terms!! CONVERSE Corollary Proof by Contradiction If and only if Axiom Implies

  2. Converse of a Statement Statement:IfRex is a dog thenRex is a mammal. (True) Converse:IfRex is a mammal thenRex is a Dog. (False)

  3. What is the Converse of a Theorem? Consider the sentence below: If If the angles of the shape add up to 180o the angles of the shape add up to 180o Then Then the shape is a triangle. the shape is a triangle. To make the converse statement swap around the parts of the statement in the green boxes above This is the converse statement (TRUE)

  4. Not all converse statements are true. Consider the sentence below: If a shape is a square. a shape is a square If Then the angles add up to 360o the angles add up to 360o Then Now make the converse statement. Can you think of a shape with angles of 360o which is not a square ? Any closed quadrilateral.

  5. Is the Converse True or False? True (1) If a triangle has three equal sides then it has three equal angles. True (2) If a number is even then the number divides by two exactly. False (3) If a shape is a square then the shape has parallel sides. False (4) If you have thrown a three and a four then your total score is seven with a die.

  6. Introducing Indirect Proof: Leinster game? Paul and Mike are driving past the Aviva Stadium. The floodlights are on. Paul:Are Leinster playing tonight? Mike:I don’t think so. If a game were being played right now we would see or hear a big crowd but the stands are empty and there isn’t any noise. Reductio ad Absurdum: Proof by Contradiction

  7. Introducing Indirect Proof Sarah left her house at 9:30 AM and arrived at her aunts house 80 miles away at 10:30 AM. Use an indirect proof to show that Sarah exceeded the 55 mph speed limit.

  8. Proof by Contradiction : Algebra

  9. Proof by Contradiction: Inequalities If Tim buys two shirts for just over €60, can you prove that at least one of the shirts cost more than €30??

  10. Assume neither shirt costs more than €30 x ≤ 30 + + y ≤ 30 x + y ≤ 60 This is a contradiction since we know Tim spent more than €60 Our original assumption must be false At least one of the shirts had to have cost more than €30 QED

  11. Geometry : Proof by Contradiction Triangle ABC has no more than one right angle. Can you complete a proof by contradiction for this statement? Assume ∠A and ∠B are right angles We know ∠A + ∠B + ∠C = 1800 By substitution 900 + 900 + ∠C = 1800 ∴ ∠C = 00 which is a contradiction ∴ ∠A and ∠B cannot both be right angles ⇒ A triangle can have at most one right angle

  12. 2 1 1 Proof by Contradiction: The Square Root of 2 is Irrational To prove that 2 is irrational Assume the contrary: 2 is rational i.e. there exists integers p and q with no common factors such that: (Square both sides) (Multiply both sides by ) (......it’s a multiple of 2) (......even2 = even)

  13. (Divide both sides by 2) . This contradicts the original assumption. 2 is irrationalQ.E.D.

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