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Unformatted text preview: 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 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 ....
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This note was uploaded on 06/12/2011 for the course MATH 103 taught by Professor Wouters during the Spring '08 term at Wisc Oshkosh.
- Spring '08