CMPSC 360: Dr.
Discrete Mathematics for CS Pennsylvania State University
Practice Midterm Exam 1
Exam Date: 2/17/2022
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Exam Time: 8 PM – 10 PM, Room Numbers: 105 FORUM
• Do not open this exam booklet until you are directed to do so. Read all the instructions on this page.
• This exam contains 5 problems, some with multiple parts. You have 110 minutes to earn 100 points. Students who is registered to SDR, get extended time.
• This exam booklet contains 12 pages, including this one.
Two extra sheets of scratch paper are attached. Please detach them before turning in your exam.
• This exam is closed book.
No calculators, mobile, smart watch or programmable devices are permitted.
• Write your solutions in the space provided. If you need more space, write on the extra sheet attached at the end of this booklet.
• Do not spend too much time on any one problem. Read them all through first, and attack them in the order that allows you to make the most progress.
• You will be graded not only on the correctness of your answer, but also on the clarity with which you express it. Be neat.
• Good luck!
Name: ID: Sec:
Problem 1 2 3 4 5 Total
Points 20 20 20 20 10 90
CMPSC 360 Exam 1
— Extra space. —
CMPSC 360 Exam 1
(a) Write each of the following sets by listing their elements between braces.
i. A={x ∈ R|x2 = 9} Answer:
ii. B={x ∈ R|x2 + 5x = 0} Answer:
(b) What is the cartesian product of A × (B × B) where set A and set B are defined in 1(a) (5pts) Answer:
(c) Write each of the following sets in set-builder notation
i. {2,5,10,17……}
ii. {2,1,0,1,2} Answer:
(d) Let p, q and r be the propositions:
x: You get an A in this class
y: You do every exercise in this
z: You get an A on the final
Write these propositions using x, y and z and logical connectives(including negations)
i. For getting an A in the class, it is necessary to get A in the final exam and doing every exercise in this book.
ii. You get an A on the final, but you don’t do every exercise in this book; never- theless, you get an A in this class.
CMPSC 360 Exam 1
2. (Logics)
(a) Using De Morgan‘s Laws, find the negation of the followings: y < −6 and y < 0
(b) Express the negations of this propositions using quantifiers
”For every student in this class, there is some exercise that he or she has not solved”. Let S(x, y) be O`student x has not solved exercise y. (2.5pts)
(c) Determine the truth value of each statements: (5pts)
i. If the domain consists of all integers, what is the truth value of ∀x(2x > x) . Draw the circle for your answer.
ii. If the domain consists of all real numbers, what is the truth value of ∃a(a4 < a2). Draw the circle for your answer.
(d) Determine whether (¬a ∧ (a → b)) → ¬b is a tautology
(Without using Truth table). (10pts.)
CMPSC 360 Exam 1
3. (Rules of Inference)
(a) Show that the argument form with premises: (p ∧ t) → (r ∨ s),
q → (u ∧ t),
Show that conclusion r is valid.
(b) Show that the premises “A student in this class has not read the Chapter 1 from the text book,” and “Everyone in this class passed the first quiz” imply the conclusion “Someone who passed the first quiz has not read the Chapter 1.” (10 pts)
CMPSC 360 Exam 1 page 9
4. (Proofs.) ”For all integers n, if n3 is even then n is even.”
(a) Prove the statement by contradiction. (10pts.)
(b) Prove the statement by contraposition. (10pts.) Answer:
CMPSC 360 Exam 1 page 11
5. Let x be a number with x > 1. Prove that √x is strictly between 1 and x.
Theorem: Logical equivalence p, q, r are variables, t= tautology, c= contradiction
Theorem And Or Commutative p∧q=q∧p p∨q=q∨p
¬(p ∨ q) = ¬p ∧ ¬q p ∨ ¬p = T
p ∨ (p ∧ q) = p
(p ∧ q) ∧ r = p ∧ (q ∧ r) (p ∨ q) ∨ r = p ∨ (q ∨ r) p∧(q∨r)=(p∧q)∨(p∧r) p∨(q∧r)=(p∨q)∧(p∨r) p∧t = p p∨c = p
p ∨ ¬p = t p ∧ ¬p = c
Associative laws
Distributive
Double Negation
Idempotent p∧p=p p∨p=p
Domination Laws De morgan Negation Laws Absorption
¬(¬p) = p p ∨ t = t
¬(p ∧ q) = ¬p ∨ ¬q p ∧ ¬p = F
p ∧ (p ∨ q) = p
Apendix:Logical Equivalence:Conditional and biconditional
Apendix:Rules of Inference:
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