1. Suppose that A is a set and {Bi | i I} is an indexed families of sets. Prove that A (UiI Bi) = UiI (A Bi). This problem is similar to Theorem 4.1.3 and to Exercise 11 in Section 4.1 of your SNHU MAT299 textbook. 2. Suppose that A = {1, 2, 3}, B = {4, 5}, C = {a, b, c, d}, R = {(1, b), (2, a), (2, b), (2, c), (3, d)} and S = {(4, a), (4, d), (5, b), (5, c)}. Note that R is a relation from A to C and S is a relation from B to C. a. Find S-1 R. This is a relation from which set to which other set? Justify your solution. b. Find R-1 S. This is a relation from which set to which other set? Justify your solution. This problem is similar to Example 4.2.4 and to Exercise 5 in Section 4.2 of your SNHU MAT299 textbook. 3. Suppose R and S are relations from A to B. Must the following statements be true? Justify your answers with proofs or counterexamples. a. R = Dom(R) Ran(R) b. (R S)-1 = R-1 S-1 This problem is similar to Example 4.2 and to Exercise 9 in Section 4.2 of your SNHU MAT299 textbook. 4. List the ordered pairs in the relations represented by the following graph. Determine whether this relation is reflexive, symmetric, or transitive. Justify your answers with reasoning or counterexamples. This problem is similar to Example 4.3.3 and to Exercise 4 in Section 4.3 of your SNHU MAT299 textbook. 5. Consider the function f: defined on positive integers with f(n) = n \"flipped\" as a mirror image into a decimal. For example, f(5) = .5, f(418) = .814, and f(1000) = .0001. Define a relation R on the positive integers as (m, n) R if and only if f(m) f(n). For example, (5, 418) R because .5 .814 but (418, .923) R because .814 > .329. Is R a partial order? Either SNHU MAT299 Page 1 of 2 Module 6 Homework provide a proof to show that this is true or provide a counterexample to show that this is false. This problem is similar to Example 4.4 and to Exercise 7 in Section 4.4 of your SNHU MAT299 textbook. 6. Define a relation R on as (a, b) R if and only if a and b, when written out, have the same number of 5s. For example, (1752, 95) R since they both have one 5 but (1752, 505) R since 1752 has one 5 but 505 has two 5s. Is R an equivalence relation? Prove that R is an equivalence relation. This problem is similar to Example 4.6 and to Exercise 16 in Section 4.6 of your SNHU MAT299 textbook. SNHU MAT299 Page 2 of 2 Module 6