Let S denote the set of all 4-element permutations of the set {1, 2, 3, 4, 5, 6, 7, 8, 9). Define an equivalence relation on S as follows: two permutations x, y € S are related if they both permute the same elements (i.e., generate the same subset). Determine the size of the equivalence class S6789 [6789] that contains the permutation = 6789. Enter your answer in the box below.

H3. Let S be the set of all binary strings. The relation R on S defined by aRb if a and b have the same length is an equivalence relation. What is the equivalence class [01]R of 01?

Let R = {(a, a),(a, b), (b, b), (a, c), (c, c)} be a partial order relation on Σ = {a,b,c}. Let be the corresponding lexicographic order on Σ*, the set of all finite strings over Σ. Which of the following is true? Select one: O a. abbac abb O b. abbac O c. O d. abbac abbaqc O e. abbaaacc abbaab be ba abbab

Find the number of different partions of a set (a) with one element (b) with two elements (c) with four elements   Prove the second part of the “Equivalence Relations and Partions” Theorem. That is, prove that given a partition of a set S the relation defined as follows “xRy if and only if x and y are in the same cell” is an equivalence relation on S.   Determine, with justification, whether the following relations are equivalence relations. If so, describe the partition arising from the equivalence relation. (a) x ∼ y in R if x ≥ y (b) n ∼ m in Z if |n| = |m|. We saw in class that the residue classes modulo 3 in Z + are {1, 4, 7, 10, . . . } {2, 5, 8, 11, . . . } {3, 6, 9, 12, . . . } Write the residue classes modulo n in Z + for (a) n = 4 (b) n = 5   Simplify the given expression and write your answer in the form a + bi for a, b ∈ R. (a) i^3 (b) i^5 (c) i^23 (d) (5 + 6i)(7 − i) (e) (1 + i)^3 (f) |5 + 3i|