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|
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