Transcript Slide 1
Purpose: Let’s Define Some Terms!! CONVERSE If and only if 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) Algebra If 3( x 2) 42 then x 16 (True) If x =16 then 3( x 2) 42 (True) What is the Converse of a Theorem? Consider the sentence below: If Then the angles of the shape add up to 180o 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) Not all converse statements are true. Consider the sentence below: If aa shape shape isis aa square. square Then the theangles anglesadd addup upto to360 360oo Now make the converse statement. Can you think of a shape with angles of 360o which is not a square ? Any closed quadrilateral. Is the Converse True or False? (1) If a triangle has three equal sides then it has True three equal angles. (2) If a number is even then the number divides True by two exactly. (3) If a shape is a square then the shape has False parallel sides. (4) If you have thrown a three and a four then False your total score is seven with a die. 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 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. Proof by Contradiction : Algebra Theorem: There is no solution to the equation x 2 3x 1 x x3 Proof: (By Contradiction) Suppose there IS a solution. Call it p p2 3 p 1 p p3 p 2 3 p 1 p ( p 3) p2 3 p 1 p2 3 p 1 0 ............Impossible Hence, no solution exists. Q.E.D. 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?? i.e. If x y 60 then either x 30 or y 30 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 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 Proof by Contradiction: The Square Root of 2 is Irrational 2 To prove that 2 is irrational Assume the contrary: 2 is rational 1 i.e. there exists integers p and q with no common factors such that: p 2 q (Square both sides) p2 2 2 q (Multiply both sides by q 2) p 2 2q 2 p 2 is even (......it’s a multiple of 2) p is even (......even2 = even) 1 p is even p 2k for some k p 2 q If p 2k p2 2 2 q p 2q 2 p 2 2q 2 becomes (2k ) 2 2q 2 2 . p is even 2 p is even 4k 2 2q 2 (Divide both sides by 2) 2k 2 q 2 Then similarly q 2m for some m p 2k p has a factor of 2 in common. q 2m q This contradicts the original assumption. 2 is irrational Q.E.D. Click for link to external website