# Convention notes notes notes 2 ez ozatla

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Notes
More Definitions Definition Supposeaandbare integers. We sayadividesb, writtena|bifb=acfor somec2Z. In this case we also say thatais adivisorofb, andbis amultipleofa.Example:10 = 2·5I2|10, 5|10I2 is a divisor of 10I10 is a multiple of 2IIs 0 a divisor of 10?No. There is noa2Zwith 0a= 10.We write 06|10IIs-1 a divisor of 10? Notes 3
Direct Proof of Conditional Statements How to prove P ) Q Proposition: If P , then Q . Proof : Suppose P . . . . Therefore Q . Every step should be completely justified. P Q P ) Q T T T T F F F T T F F T Direct Proof Prove the following: If 4 | x , then x 2 is even. Proposition: If 4 | x , then x 2 is even. Proof : Suppose 4 | x . Then by the definition of “divides,” there exists some b 2 Z such that 4 b = x . Then x 2 = 4 b 2 = 2 b Therefore, x 2 = 2 ab for some ab 2 Z . Therefore, x 2 is even by the definition of “even”. Then there exists some a 2 Z such that 4 a = x . Then x 2 = 2 a , so x 2 is even. Direct Proof Prove the following: If x is odd, then x 2 + 1 is even. Proposition: If x is odd, then x 2 + 1 is even. Proof : Suppose x is odd. Then x = 2 a + 1 for some integer a . Then x 2 +1 = (2 a +1) 2 +1 = 4 a 2 +4 a +1+1 = 2(2 a 2 +2 a +1).

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• Fall '19
• Negative and non-negative numbers, Prime number, Divisor, Parity, Evenness of zero