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CMSC 150 – INTRODUCTION TO DISCRETE STRUCTURES FINAL EXAMINATION 1. (10 PTS) FOR EACH THE FOLLOWING GROUPS OF SETS, DETERMINE WHETHER THEY FORM A PARTITION FOR THE SET OF INTEGERS. EXPLAIN YOUR ANSWER.

CMSC 150 – Introduction to Discrete StructuresFinal Examination1. (10 pts) For each the following groups of sets, determine whether they form apartition for the set of integers. Explain your answer.a.A1 = {n Z : n > 0}A2 = {n Z : n < 0}b.B1 = {n Z : n = 2k, for some integer k}B2 = {n Z : n = 2k + 1, for some integer k}B3 = {n Z : n = 3k, for some integer k}2. (10 pts) Define f: Z Z by the rule f(x) = 6x + 1, for all integers x.a.Is f onto?b.Is f one-to-one?c.Is it a one-to-one correspondence?d.Find the range of f.Explain each of your answers.3. (10 pts) f: R R and g: R R are defined by the rules:f(x) = x2 + 2 x Rg(y) = 2y + 3 y RFind f ◦ g and g◦ f.4. (10 pts) Determine whether the following binary relations are reflexive,symmetric, antisymmetric and transitive:a.x R y xy ≥ 0 x, y Rb.x R y x > y x, y Rc.x R y |x| = |y| x, y RFor each of the above, indicate whether it is an equivalence relation or a partialorder. If it is a partial order, indicate whether it is a total order. If it is anequivalence relation, describe its equivalence classes.5. (10 pts) Determine whether the following pair of statements are logicallyequivalent. Justify your answer using a truth table.p (q r)andpqr6. (10 pts) Prove or disprove the following statement:n ,m Z, If n is even and m is odd, then n + m is oddThen write the negation of this statement and prove or disprove it.7. (10 pts) Prove the following by induction:n3i – 2 =i=132 −28. (10 pts) Use the permutation formula to calculate the number permutations ofthe set {V, W, X, Y, Z} taken three at a time. Also list these permutations.9. (10 pts) Translate the following English sentences into statements of predicatecalculus that contain double quantifiers and explain whether it is a truestatement.a.Every rational number is the reciprocal of some other rationalnumber.

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