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Math 260 Final Exam Name___________________ I. Directions: For each statement write “True” or “False” in the column to the right. If an answer is false, provide a counterexample or correction, where applicable. (20 pts) Statement True or False Possible Counterexample or Correction 1. P <=> Q means “P iff Q.” 2. Let S = {1,2,3,4,5,6,7,8,9,10}. Then, the power set Р(S) has 1000 elements. 3. Uc = { }, where U is the Universal set. 4. (PQ) <=> (~P ~Q) 5. h(x)= 1/sin(x) is a well-defined function on domain . 6. A, where A is any set and Φ={ }. 7. Suppose set D is both domain and codomain for the relation r = {(x,x): x D }; then r is an equivalence relation on D. 8. When defined on domain , f(x) = 10x is onto . 9. When defined on , the function g(x)= x3-x is 1-1 on its range. 10. {x|x=4m, m }={y| y=8n, n } II. Prove or Disprove: P↔Q (P Q) /\ (PQ) . (10 pts) III. Perform the following operations. Show work to get credit! (10 pts) A. Change 78910 to a base 5 number. B. 110000112 - 111112 Math 260 Final Exam Name___________________ IV. State and demonstrate three forms of the Fundamental Theorem of Algebra, using the polynomial f(x) = 4x2-4x-3. (12 pts) V. For the following, suppose U = {1,2,3,4,5,6,7,8,9,10}, S = {3,4}, T = {2,3,5,8,9}, and Φ = { }. Write the resulting sets using list notation. (Note: U is the universal set of discourse.) (20 pts) 1. (U S) TC = 2. S X T = 3. UC = 4. (S T)c = 5. (U \ T ) \ S = 6. List below the Power Set of T \ S. 7. List below a set of ordered pairs, representing a 1-1 function f with domain S and codomain T. 8. List below a set of ordered pairs representing a function g that is NOT onto, with domain T and codomain S. 9. List below a set of ordered pairs, representing an ONTO function h with domain T and codomain S. 10. List the LARGEST equivalence relation ~ on S. Math 260 VI. For the following, let x, y Final Exam Name___________________ . Define x ~y <=> xmod4 = ymod4. (15 pts) A. Prove that ~ is an equivalence relation on . B. Describe the equivalence classes for ~ using proper set notation. C. Prove or disprove: If xmod4 = 3 and ymod4 = 2, then (xy)mod4 = 2. VII. Prove that .787878… is rational by finding its rational equivalent. You must show your work! (3 pts) Math 260 Final Exam Name___________________ VIII. Prove or disprove the following. (10 pts) A. Let S = {x x , 6 x} and T = {y y , 3 y}. Prove or disprove that S T. B. Let S = {x x , 6 x} and T = {y y , 3 y}. Prove or disprove that T S. C. Prove that if x2 + y2 is even , then x + y is even. Math 260 Final Exam Name___________________