By V. A. Krechmar

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Dependence on the axioms is not spelled out, but the tone and style suggest that the writer understands what is going on, that (s)he could fill in whatever details have been omitted if challenged, and that the writer believes the key ideas have been conveyed to the (qualified) reader. But what if the reader does not understand how the steps in the proof are justified? Would you be able to spell out missing details when challenged? Test yourself. Template for Proofs By Induction. Formulate P(k).

4 The Binomial Theorem 41 Proof. We prove this by induction on k ≥ 0. The base case follows with x0 = 1 (which we use for x = a + b, a, and b) and 00 = 1. For the induction step, assume that (a + b)n = ∑nm=0 Then n+1 ∑ m=0 n m am bn−m for some n ≥ 0. 17 with r = 1. 42 4 Recursion = (a + b) (a + b)n (by distributivity) = (a + b)n+1 (by definition of powers). 22. For k ∈ Z≥0 , ∑ m=0 k = 2k . m A slight variation of the binomial theorem is the general product formula of calculus— although this looks like a completely different topic at first sight.

Segment; (iii) negate the final statement. 4 does not hold” can be written as (∃ m ∈ Z such that) (∀ n ∈ Z) m + n = 0 . 7. Negate the following statements. Note that you do not need to know the meaning of these statements in order to negate them. (i) Every cubic polynomial has a real root. (ii) G is normal and H is regular. (iii) ∃! 0 such that ∀ x, x + 0 = x. (iv) The newspaper article was neither accurate nor entertaining. (v) If gcd(m, n) is odd, then m or n is odd. (vi) H/N is a normal subgroup of G/N if and only if H is a normal subgroup of G.

### A Problem Book in Algebra by V. A. Krechmar

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