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Exams are to be done individually and must represent original work—it is a violation of the honor code to copy or derive exam question solutions ...
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Exams are to be done individually and must represent original work—it is a violation of the honor code to copy or derive exam question solutions from other students or anyone at all, textbooks, or previous instances of this course.
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Exercise 1 (10 points). Prove that all odd perfect squares are congruent to 1 modulo 4.
Exercise 2 (10 points). Consider a relation ∝ on the set of functions from N+^ to R, such that f ∝ g if and only if f = O(g). Is ∝ an equivalence relation? A partial order? A total order? Prove.
Exercise 4 (10 points). The drama club has m members and the dance club has n members. For an upcoming musical, a committee of k people needs to be formed with at least one member from each club. If the clubs have exactly r members in common, what is the number of ways the committee may be chosen? Substantiate.
Exercise 5 (10 points). How many nonnegative integers less than or equal to 300 are coprime with 144? Substantiate.
Exercise 7 (10 points). You already know from Bezout’s Identity that if a and b are coprime integers, then there are integers x and y such that ax + by = 1. Now prove the same result using the Pigeonhole Principle. (You may assume that a and b are positive.) Hint: Take the remainders, modulo b, of the first b − 1 positive multiples of a, and consider what happens if 1 is not in this set.
Exercise 8 (10 points). Prove that at a cocktail party with ten or more people, there are either three mutual acquaintances or four mutual strangers.
Exercise 10 (10 points). Let G be a graph that has no induced subgraphs that are P 4 or C 3. Prove that G is bipartite.