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lecture04_proofs

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Methods of Proof Mariusz Bajger COMP2781/8781 School of Computer Science, Engineering and Mathematics March 21, 2011 1 / 20 Reading and Exercises Reading Epp, Chapter 4 Exercises All blue ones for each of the sections. Good learning strategy: BE ACTIVE! I Regularly revise lectures I Solve the suggested exercises I Be critical when reading textbook/lectures I Ask your colleagues, ask the lecturer, don’t be shy! I It is OK to ask for help any time 2 / 20 Assumptions and definitions I Basic algebraic properties of R , Q , Z are assumed (see Appendix A) I Equality properties: a = a , if a = b then b = a , if a = b and b = c then a = c I Z is closed under addition/subtraction and multiplication (but not division) Definition n Z is even n = 2 k , for some integer k . n Z is odd n = 2 p + 1, for some integer p . p Z is prime p > 1, and for all r , s Z + , if p = rs , then either r or s equals p (exclusive or). n Z is composite ⇔ ∃ r , s Z + : n = rs and r , s (0 , n ). Note: every integer n > 1 is either prime or composite . (See Epp. Ex. 4.1.2). 3 / 20 Rational numbers Quotients of integers are mostly not integers. We call them rationals. Definition A real number r is rational if, and only if, a , b Z such that r = a b , and b ̸ = 0. I Is 0 . 32323232 . . . a rational number? I Is sum, product, ratio of rationals also rational? Definition A real number that is not rational is called irrational . 4 / 20

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Proving/disproving existential statements Existential statement x D : Q ( x ) I constructive proof will find such x I nonconstructive proof will only show that such x must exist EXAMPLE (constructive proof) Let a , b R . c R : a < c < b . EXAMPLE (nonconstructive proof) Prove that x , y R \ Q such that x y Q .
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