Unformatted text preview: a , b , c ∈ R . Prove  ac  ≤  ab  +  bc  . 5 . [2 + 6 = 8 pts] (i) State the First Principle of Mathematical Induction . (ii) Prove 1 · 2 + 2 · 3 + 3 · 4 + ··· + n ( n + 1) = 1 3 n ( n + 1)( n + 2) , for all integers n ≥ 1. 6 . [2 + 5 = 7 pts] (i) Deﬁne the term ideal . (ii) Let a ∈ Z . Prove that the set J a := { ka : k ∈ Z } is an ideal of Z . 7 . [2 + 6 = 8 pts] (i) Deﬁne what it means for two integers to be relatively prime . (ii) Let a , b ∈ Z . Prove that gcd( a,b ) = 1 if and only if gcd( a + b,ab ) = 1. 1...
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 Spring '08
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 Natural number, Prime number, Mathematical logic, Rational number, R. Prove

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