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Math 61 Practice First Test

# Math 61 Practice First Test - Practice First Test...

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Practice First Test Mathematics 61 Disclaimer: Listed here is a selection of the many possible sorts of problems. The actual test is apt to make a different selection. 1. Let S5 be the set of all binary strings of length 5 (such as 11001). Let E be the equivalence relation on S5 for which sEt if and only if s and t have the same first three bits. (For example, 11011 E 11010.) (a) Find [10101], the equivalence class of the string 10101. (b) How many equivalence classes are there altogether? (Support your answer.) (c) Let A = {4, 5, 6, 7} and let P = {{5, 7}, {4, 6}}. Then P is a partition of A. Give the matrix (relative to numerical order on A) of the equivalence relation R on A for which A/R = P. 2. (a) Assume that R is a transitive relation. Show that whenever (x, y) is in R o R, then (x, y) is also in R. (b) Assume that Q is a relation and that Q o Q is a subset of Q. Prove that Q is transitive. 3. Let f(n) = sqrt(4n), and let g(n) = lg 4n. Determine whether f is O(g) and whether g is O(f). (Notation: sqrt(w) is the square root of w. And lg n is the logarithm of n to the base 2.) 4. Assume that E is an equivalence relation on a set S.

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