# STEVENS MA 334B - MA 334 Final Exam (16 pages)

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## MA 334 Final Exam

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Lecture Notes

- Pages:
- 16
- School:
- Stevens Institute of Technology
- Course:
- Ma 334b - Discrete Mathematics

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Name MA334 SSN FINAL EXAM I pledge my honor that I have abided by the Stevens Honor System 1 10pts Use a truth table to check the validity of the following argument P1 If Bob is not on the Mets then Tom is on the Yankees P2 If Tom is not on the Yankees then Bob is on the Mets C Bob is not on the Mets or Tom is not on the Yankees Grade December 1999 2 8pts Let U Z Are the following true or false Explain x y x y 0 y x x y 0 x y xy 0 x y xy 1 3 10pts Prove or disprove The even negative integers are a countably infinite set 4 10pts Prove that for all integers n n is odd if and only if n3 is odd Note Do not use induction 5 14pts Let A B C be subsets of a universe U Prove using the element method or disprove with a counterexample 1 If C A B then B C 2 C B A A B C 3 A x B C AxB AxC 6 12pts Partition the set of real numbers R into 5 sets in such a way that the integers 1 2 and 3 fall into the same set in the partition as 2 Let T 5 6 7 16 If 7 integers are chosen from T show that some two of them must sum to 21 Explain 7 12pts 1 Find a set A and a 1 1 function f A A that is not an onto function 2 Let f A B and g B C be onto functions Prove that gof A C is an onto function 8 12pts Let 0 1 and A 5 For s t A let sRt if and only if the first three characters of s are the same as the first three characters of t Show that R is an equivalence relation What are the equivalence classes of R 9 12pts Prove or disprove If f N N given by f n 5n 3 is a 1 1 function Let A x R 5 x 5 and B x R 0 x 10 Sketch A B R 0 1 10 12pts Decide if each of the following relations on the given set A is reflexive symmetric antisymmetric and or transitive If not explain 1 A where 0 1 R1 w1 w2 w1 w2 has an even number of 1 s 2 A a b c d e R2 a a b a a e a d b c c d d c d a 3 A Z R3 x y x y 2 11 12pts 1 Let A a b c d Find an antisymmetric relation R1 on A and a different antisymmetric relation R2 on A such that R1 R2 is not antisymmetric 2 Let R be a relation on the set of points in the plane given by x1 y1 R

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