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Unformatted text preview: 1 MA1100 Lecture 8 Mathematical Proofs • Uniqueness Statements • Equivalence • True or False • Other Remarks • Common Errors MA1100 Lecture 8 2 More than an Existence statement Proposition For a, b, c œ R , if a ∫ 0 and b 2 = 4ac, then the quadratic equation ax 2 + bx + c = 0 has exactly one real number solution. Two things to prove: (1) There exists a real solution for the equation. (2) The solution in part (1) is the only solution. MA1100 Lecture 8 3 More than an Existence statement Proof We first show there exists a real solution for ax 2 + bx + c = 0 . Suppose a ∫ 0 and b 2 = 4ac. a 2 b x − = We claim that is a solution. Substituting this in the left side of the equation, we have c a 2 b b a 2 b a 2 + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − Since b 2 = 4ac, so the above expression is 0. Hence the equation is satisfied and we have shown that x = b/2a is a solution. 2 2 a 4 ac 4 b + − = Constructive proof MA1100 Lecture 8 4 More than an Existence statement Proof (cont.) Next we show this is the only solution for ax 2 + bx + c = 0 . Suppose a ∫ 0 and b 2 = 4ac. Since b 2 = 4ac, so (2) becomes Multiplying 4a to (1): 4a 2 r 2 + 4abr + 4ac = 0 (2) 4a 2 r 2 + 4abr + b 2 = 0 By factorizing the left side, we get (2ar + b) 2 = 0 This implies (2ar + b) = 0 This shows that r=b/2a is the only solution. a 2 b r gives which − = Suppose r is a solution of the equation: ar 2 + br + c = 0 (1) MA1100 Lecture 8 5 Uniqueness Statements 1. The equation x 3 + 1 = 0 has a unique real solution. 2. There is exactly one angle q with 0 § q § p such that cos( q ) = q . 3. There is only one prime number of the form n 31 for some integer n. Example Keywords Exactly one, unique, only one Uniqueness statements claim that an object that satisfies certain property is unique . i.e. there is only one such object. MA1100 Lecture 8 6 Uniqueness Statements If two objects a and b both satisfy condition P(x), then a and b are the same. ( " a) ( " b){ [P(a) ⁄ P(b)] Ø [a = b] } Symbolic form There is exactly one object x that satisfies property P(x). ( $ x) P(x) ( $ ! x) P(x) Existence part Uniqueness part MA1100 Lecture 8 7 Uniqueness Statements 1. Direct proof: Assume a and b both satisfy property P(x)....
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This note was uploaded on 03/19/2012 for the course SCIENCE MA1100 taught by Professor Forgot during the Fall '08 term at National University of Singapore.
 Fall '08
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