MATH 263 Towson University Discrete Mathematics Exam This exam covers the material from Chapter 6, 7 and 8 that we studied in this course. The sections are | Course Hero
MATH 263 Towson University Discrete Mathematics Exam This exam covers the material from Chapter 6, 7 and 8 that we studied in this course. The sections are 6.1 — 6.3; 7.1 — 7.3 and 8.1 — 8.3. You need to understand sets and set operations, functions and the onto and one-to-one properties, bijections, binary relations, the reflexive, symmetric and transitive properties, and equivalence relations. Math 263 Discrete Mathematics
Dr Goode
It has been a privilege to teach you. Enjoy the rest of your summer!
Exam #3
100 pts
(0) Create a cover page for this exam that is a whole piece of paper with the following written on it:
Your First and Last Name
Your original Section Number for this course (Sec 020 or Sec 021)
Exam #3
I affirm that I have neither given nor received any unauthorized assistance on this exam.
SIGN and DATE the academic integrity statement above.
Name your .pdf file as follows:LastName.FirstName.Exam3.Sec02x where either x = 0 or x = 1, depending upon which
section you originally enrolled in.
Number your problems as indicated. Skip at least one line between consecutive problems and skip a line between
consecutive problem parts. Make sure your problems and pages are in order.
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You must work this Exam ON YOUR OWN. DO NOT look on the internet for answers. Do not confer with any other student
about the questions on this exam. If you are aware of posted solutions that are available to students in this course and do
not report that, then you are cheating. Students who cheat will be prosecuted according to guidelines set out by TUs Office
of Student Conduct and Civility Education.
Use ONLY the NOTATION SEEN IN LECTURES and in the TEXTBOOK.
I apologize for having to state the obvious to those of you who do not consider cheating a good way to get a degree.
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Use only the definitions for any proofs that you write for this exam. Do not use any prior results. Indicate where your
proofs and counterexamples begin. Use grammatically correct English and proper capitalization and punctuation. Leave
NOTHING to my imagination. If you wonder whether you should write something, you probably should write it. Your reader
should not need to perform any calculations or make any inferences.
(1) (3pts ea) Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3}, B = {4, 5, 6, 7, 8} and C = {4, 6, 8}. Find each set, i.e., give the
elements in each set. Use set brackets because the answers are sets.
(a) B ? (A ? C)c .
(b) (A ? B) ? (B ? A).
(c) A × C.
(d) P(C).
(2) (4pts ea) Let X = {x, y, z} and Y = {m, n}.
(a) Construct a function f from X to X that is not bijective if it is possible to do so. In that case, verify that f is a
function, and explain exactly why it is not bijective. If it is not possible to construct such a function, explain exactly why it
is not possible.
(b) Construct a function g from X to Y that is bijective, if it is possible to do so. In that case, verify that g and explain
exactly why it is a bijection. If it is not possible to construct such a function, explain exactly why it is not possible.
(3) (4pts ea) Let A = {1, 2, 3, 4}.
(a) Construct a relation on A that is reflexive, but not symmetric and not transitive.
(b) Construct a relation on A that is not reflexive, but is symmetric and is transitive.
(4) Let S be the set of all binary strings, i.e., the set of all strings written using 00 s and 10 s. Define a binary relation R on S
as follows: For all strings s and w in S, sRw if and only if s and w begin with the same digit.
(a) (6pts) Is R an equivalence relation in S? Justify your response.
(b) (6pts) Is R a function from S to S? Justify your response.
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For Problems #5 #9, you willl either be asked to prove a statement or disprove a statement, or decide if a statement
is true or false, then prove or disprove the statement. Prove statements using only the definitions. DO NOT use any set
identities or any prior results whatsoever. Disprove false statements by giving counterexample and explaining precisely why
your counterexample disproves the claim.
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(5) (12pts) Consider the < relation defined on R as usual, where x < y if and only if x represents a point on the number line
that is to the left of the point that y represents. Prove if true, and disprove with a counterexample if false: The < relation is
neither reflexive nor symmetric, but it is transitive.
(6) (12pts) Prove if true and disprove if false, using NO prior results:
For all subsets A, B, C of a common universe U, (A ? B)c ? C = Ac ? B c ? C.
(7) (12pts) Let A be any set with at least two elements. Consider the relation R defined on P(A) defined as follows:
?B, C ? P(A), BRC ? B ? C 6= ?. For each of the three properties Reflexivity, Symmetry and Transitivity, determine
whther or not this relation has the property. For each property either prove that R has the property, or show why it does
not have the property with a counterexample. Be thorough.
(8) (12pts) Prove if true and disprove if false, using NO prior results: For all sets A, B and C, if B ? A and C ? B, then
(C ? A) ? (B ? A) = ?.
(9) (12pts) Prove if true, and disprove if false, using NO prior results: ? sets A and B, P(A ? B) = P(A) ? P(B).
EXTRA CREDIT: Let A = {1, 2, 3, 4}. Is it possible to build a binary relation on A that is also a function from A to A?
If so, construct such a function and explain why it is a function and why it is an equivalence relation. If it is not possible,
explain why it is not possible.
End Exam #3
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