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1. Suppose that A is a set and {Bi ie |} is an indexed families of sets. Prove that A x (Un Bi) = Uid (A x Bi). 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 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. 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. (RnS)¹ R-inS-1 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. a b c d 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) C R if and only if f(m) < f(n) For example, (5, 418) e R because 5 5 814 but (418, 923) c R because 814 > 329. Is R a partial order? Either provide a proof to show that this is true or provide a counterexample to show that this is false 6. Define a relation R on Z as (a, b) e R if and only if a and b, when written out, have the same number of 5s. For example, (1752, 95) e R since they both have one 5 but (1752, 505) e R since 1752 has one 5 but 505 has two 5s. Is R an equivalence relation? Prove that R is an equivalence relation. 7. Suppose that x²y + xy² + y³ = x³: a) Using the method of proof by contrapositive, show that if x and y are not both zero, then 0. b) Using the method of proof by contradiction, show that if x and y are not both zero, then 0. 8. Prove the following two statements: a) For every integer n, 72 in iff 8| n and 9 n. b) It is not true that for every integer n, 90 In iff 6 in and 15 In. 9. Let a, b, and cbe real numbers with a 0. Prove that lime-e (ax b) =ac+ b. 10. Suppose that {Ail icl} is an indexed family of sets and B is a set. Prove that (niel Ai) x B = Qiel (Ai B). 11. Suppose R is a partial order on A and S is a partial order on B. Define a relation T on A x B such that (a1, b1) T (a2, b2) iff a1 R a2 and b1 is b2. Is T a partial order on A B? Either provide a proof to show that this is true or provide a counterexample to show that this is false. 12. Prove that if x #3, then there exists a real number such that x - (3y 2), (y ).(Please Explain this mathematically and with words as detailed as possible as to the steps you took to solve this.

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