COMS W3203 Study Guide - Midterm Guide: Mathematical Induction, Bijection, Surjective Function
19 Dec 2017
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If something is taking too long, move on to the next question. Note that this is a sample exam and while it bears some similarity with the real exam, the two are not isomorphic. Prove the following by induction: state p (n), the base case, inductive hypothesis and inductive step explicitly. 4: recall n! is de ned as n (n 1) 3 2 1. N n such that n 4 n! > 2n: prove for all n n: (cosx + isinx)n = cos(nx) + isin(nx) (1) (2) (3) Hint: you may use the angle sum/difference identity cos(a b) = cos(a)cos(b) sin(a)sin(b) and sin(a b) = sin(a)cos(b) + cos(a)sin(b). Let r be the set of real numbers and z be the set of integers. De ne a relation x y on r denoted xry. X, y r: x y z (4) Recall that an equivalence relation is one that is re exive, symmetric and transitive.
