MACT 3224/317: Exam N 1
Yahia El Horbaty, Khouzeima Moutanabbir, Abd-Elnasser Saad and Noha Youssef
Fall 2016
Name:
Student ID:
Section number:
Instructions
There are seven (7) written-answer questions here.
Give clear answers and explanations, justi
MACT 200 Final AUC May 31, 2011
Problem 1: (10 pts) Is the following argument valid? Give a formal
and precise proof of your answer.
Every MACT 200 student is majoring in Math or CS (or both).
Aly is taking MACT zoo.
Aly is not majoring in CS.
Therefore,
MACT 2131 Final AUC May 22, 2015
Problem 1: Consider the following predicate, where the universe is
the set Z of all integers:
S(x, y, z): x + y = z.
a) Express each of the following as a predicate statement using
the predicate S:
(.3 (D
IL
i. (2 pts) 2
MACT 2131
Solution of the Final Exam
Spring, 2016
Problem 1: Consider the following predicate, where the universe is
the set of all positive integers:
G(x, y, z): z is the greatest common divisor of x and y,
Express each of the following as a predicate st
MACT 2131
Midterm 2 Solutions
Fall 2016
Problem 1: Let = 22016, = 123456789. Find (with proof):
i)
(4 pts) lcm(, )
Solution: (1+2+1 pts)
We first note that the only prime divisor of is 2 while is odd.
Thus, gcd(, ) = 1.
Consequently, lcm(, ) =
= (12345
Midterm 1 Solutions
Problem 1: (5 pts) Is the following propositional statement a tautology, a
contradiction, or a contingency?
(p (q r) (p q) r)
This statement is a tautology. (1 pt)
Reason 1: We can construct a truth table (with 8 rows) (2 pts), and che
MACT 2131
Midterm 1 Solutions
Fall 2016
Problem 1: (5 pts) Does the exclusive or operation satisfy the
following associativity law? (Circle one and state your reason.)
(p q) r) (p (q r)
TRUE
Reason: Constructing the truth table with columns for both
sides
MALT 200 Final Dec 16, 2011
Problem 1: Consider the following two predicates, where the
universe is the set of all AUC students:
P(x): Student x is taking MACT zoo.
Q(x, y): Students x and y are friends.
Express each of the following as a predicate statem
MACT 2131 Final AUC May 24, 2014
Problem 1: Consider the following argument:
Every student is smart.
Some students are hard working.
Every smart hard working student will get an A in MACT 2131.
Therefore, some student will get an A in MACT 2131.
Midterm 2 Solutions
Problem 1: a) (3 pts) Express the number 10! = 109821 as a product
of powers of distinct primes.
Solution: (1+1+1 pts)
10! = 10987654321 = (25)(32)(23)7(23)5(22)32 = 2834527
b) (4 pts) Find (with proof) gcd(5555, 3333).
Solution 1: (1+
MACT 2131
Solution of the Final Exam
Fall, 2016
Problem 1: Consider the following predicate, where the universe is
the set of all people:
P(x, y): x is the father of y.
Express each of the following as a predicate statement:
a) (3 pts) The father of Adel
MACT 2131 ' I - ' Midterm 2 Nov 19, 2014
Problem 1: a) Let a = 5183 and b = 2599. Find (Show all work).
i) (4 pts) gcd(a, b) -.-_- 30.1 (25qu 5133 mod 2F9?)=Jca1(2fm,zr
W;
5183 0d 23% . cfw_183, 983 54? (1 : 5m (15/ ;
m (-3.5%?) . C/ Q).
3': SP'l'3.3- l-Z
MACT 213] Midterm 2 Nov 20, 2014
Page 2 MACT 2131 Midterm 2 Nov 20, 2014
Problem 2: True or False? Circle one and give your reason.
a) (4 pts) If f: R -> R and g: R > R are two onto functions, then the
function f o g must also be onto. ,
Reason: F
MACT 2131 Final Exam Dec 17, 2014
Problem 1: Consider the following predicate, where the universe is
the set of humans:
P(x, y): x is a parent of y.
Express each of the following-as a predicate statement (use the
7
symbols u and i denoting you an
MACT 2131 Midterm 1 Mar 14, 2015
Problem 1: (5 pts) Does the biconditional () satisfy the following
associativity law? (Circle one and state your reason.)
(p<>q)<>r)E(p<>(q<>r)
Reason: (Dx/I TRUE FALSE
Salami, [3 1 T FHl (Fever mr rerign)
Tm 0 o o
MACT 2131 Midterm 2 May 4, 2015
Problem 1: An mm is matrix A = [aij] is called a diagonal matrix iff
aU = 0 whenever i #j,
i.e. all off diagonal entries are zeros.
a) (3 pts) Of the following, circle those matrices that are diagonal (no
reason is required
MACT 213] Midterm 1 Oct 15, 2015
Problem 1: (5 pts) Does the conditional connective > satisfy the
following associativity law? (Circle one and state your reason.)
(P>Q)>I)E(P>(q->r)
Reasoni '- TRUE ( FALSE ,3
, b
F 1' r l9*-71,(f>~71)-yr qar 107($rr)
-V0
MACT 2131 Final Exam Dec 17, 2015
Problem 1: Consider the following predicates, where the universe
is the set Z+ of positive integers:
L(x, y): x s y;
D(x, y): x divides y;
a) (3 pts, each) Express each of the following as a predicate
statement using logi
Sc, mum x Emu; 13
MACT 2131 Midterm 1 Oct 16, 2014
Problem 1: (5 pts) Does the biconditional operation "<> satisfy the
following associativity law? (Circle one and state your reason.)
(peq)+>r)5(pe>(q<->r)
Reason: TRUE FALSE MACT 2131 Midterm 1 Oct
Nested Quantifiers
Section 1.5
1
Lecture 4 Objectives
Translate sentences involving nested quantifiers back
and forth from English to logical symbols
Find the truth value of a statement involving nested
quantifiers
Decide (with proof) if two statements
MACT 2131: Discrete Mathematics
Instructor: Wafik Lotfallah
1
Course Material
Main Text:
Discrete Mathematics and its Applications;
7th ed., 2012;
by Kenneth H. Rosen;
McGraw-Hill International Edition
+ Supplementary Material on the Blackboard.
2
Prere
MACT 2131 Homework 7 Solution
Section 2.6
2. b) [0.5 mark] Find
[
where
]
[
[
4. b) [0.5 mark] Find the product
]
[
where
]
[
[
5. [1.5 marks] Find a matrix
Let
[
such that [
] Then [
]
]
]
]
[
]
[
equations:
]
]
[
] and we have the following
, which give
MACT 2131 Homework 1 Solution
Section 1.1
2. [3 marks] Which of these are propositions? What are the truth values of those that are
propositions?
a)
b)
c)
d)
Do not pass go.
What time is it?
There are no black flies in Maine.
4 + = 5.
e) The moon is made
MACT 2131 Homework 2 Solution
Section 1.4
6. [1.5 mark] N(x):= x has visited North Dakota, domain = students in your school.
a) () = There is someone in our school who visited North Dakota
e) () = Not everyone in our school has visited North Dakota
f) ()
MACT 2131 Homework 3 Solution
Section 1.6
2. [1 mark] If George does not have eight legs, then he is not a spider.
George is a spider.
_
George has eight legs.
Let p = George does not have eight legs and q = George is not a spider This
argument has the fo
Propositional Equivalences
Section 1.3
1
Lecture 2 Objectives
Decide (with proof) if a proposition is a tautology, a
contradiction, or a contingency
Find the converse, contrapositive, and inverse of a
conditional proposition.
Decide (with proof) if two
MACT 2131 Homework 6 Solution
Section 5.1
4. [3 marks] Let
integer .
a)
b)
c)
d)
e)
be the statement that
for the positive
What is the statement
?
Show that
is true completing the basis step of the proof.
What is the inductive hypothesis?
What do you need
MACT 2131 Homework 8 Solution
Section 5.2
2. [1.5 marks] Use strong induction to show that all dominoes fall in an infinite arrangement of
dominoes if you know that the first three dominoes fall, and that when a domino falls, the domino three
farther down