Exam #1
MTH 180 - Section 0002, Spring 2021
Name:
This take-home exam is due Thursday, 3/04 at 11:00pm in Brightspace.
No exceptions!!
Before starting the exam, please read the following policies:
(1) Please print out or download the exam and complete all exam problems on the exam itself. When
you are ready to submit the exam, please do so in the Exam 1 Assignment in Brightspace and upload
your submission as a single pdf document. Brightspace will only let you submit your exam in a single
document, so make sure you are able to compile your work into one document.
(2) You may use the textbook, class notes, class slides, homework, and resources on the course site on
Brightspace as you work on the exam.
(3) You may meet with me on a one-on-one basis to ask questions on the exam. I am more than happy to
provide some hints and help, but will not walk through the exam with you. It is up to you to reach out
to me if you would like to meet regarding the exam.
(4) You may NOT work with classmates on the exam at any point and you may NOT discuss these problems
at all with classmates. This is considered cheating.
(5) You may NOT use the Internet for help on the exam problems (obviously, you may use the Internet to
access your online textbook, homework, and items on Brightspace). This is considered cheating. You
may NOT use the Internet for help on the exam problems (obviously, not including the online textbook).
This is considered cheating.
Be sure to show ALL work where appropriate. You will lose points if you simply state
answers with NO work shown.
Good luck! :)
Before submitting your exam, please sign with your name below.
Armani Fernandez
I,
acknowledge that I have read through all of the
exam policies. I understand that cheating results in an exam grade of zero with no exceptions. I understand
that cheating includes, but is not limited to, any of the following: using the Internet for help on the exam
problems, working with classmates on the exam problems, and/or discussing the exam problems with anyone
except the instructor.
1. (10 points) Answer TRUE or FALSE for each of (a)-(e). No justification is required for this problem.
(a) {5} ⊆ {2, 5, 6, 9}
TRUE
(b) 5 ⊆ {2, 5, 6, 9}
False
(c) 3 ∈ {{−1}, {3}, {8}}
FALSE
(d) {8} ∈ {{−1}, {3}, {8}}
TRUE
(e) {4, {4},
2
4 } ⊆ {4, {4}, {{4}}}
FALSE
2. (10 points) Answer TRUE or FALSE for each of the following and JUSTIFY YOUR ANSWER.
(a) {−3, −2, −1, 0, 1} = {x ∈ R | − 3 ≤ x ≤ 1}
FALSE
(b) If A = {x ∈ R | − 3 < x < 0}, B = {y ∈ Z | 0 ≤ y ≤ 2}, and C = {z ∈ R | − 3 < z < 2}, is
(A ∪ B) ⊆ C?
FALSE
3. (10 points) Let A = {−2, −1, 3} and B = {−4, −3, 6, 10}. Define a relation from A to B as follows:
For all (x, y) ∈ A × B, (x, y) ∈ R means that x2 > 10 − y.
(a) Write R as a list of ordered pairs.
AXB: {(-2,-4),(-2,-3),(-2,6),(-2,10),(-1,-4),(-1,-3),(-1,6)(-1,10),(3,-4),(3,-3),(-3,6),(3,10)
}
R = {(-2,10),(-1,10),(3,6),(3/10)
(b) Draw an arrow diagram for R.
A
B
-2
10
-1
3
6
-2 CONNECTS TO 10
-1 CONNECTS TO 10
3 CONNECTS TO 10
3 ALSO CONNECTS TO 6
(c) Is R a function? Justify your answer.
EVERY ELEMENT IN A IS NOT THE SAME AS EVERY ELEMENT INSIDE B SO NO R
IS NOT A FUNCTION.
4. (10 points) Let X, Y , and Z be subsets of the universal set U = {n ∈ Z | 0 ≤ n ≤ 10} given by
X = {n ∈ U | n = 2k for some k ∈ {1, 2, 3, 4, 5}}, Y = {0, 1, 2, 3, 7, 9}, and Z = {x ∈ U | 3 < x < 8}.
Find the following and show all of your work:
(a) X ∪ Z
XUZ{2,4,6,8,10,5,7}
X = {2,4,6,8,10}
Z{4,5,6,7}
(b) (X ∪ Y ) ∩ Z
{2,4,6,8,10} = {4,6}
(c) (Y ∪ Z)c
{0,1,2,3,4,5,6,7,9} = (8,10)
(d) (X ∪ Z) − Y
{2,4,5,6,7,8,10}
{4,6,8,5,8,10}
(e) Is {X, Y } a partition of the set U ? Justify your answer.
X,Y IS NOT A PARTITION OF THE SET U BECUASE X AND Y DO NOT
CONTAIN PARTS OF U
5. (10 points)
i. Decide the truth values of the following statement: (p ∧ ∼ q) =⇒ (∼ p ∨ q) by filling in
the truth table below. Hint: Your truth table should include seven columns including the ones
given.
p q
T T
T F
F T
F F
F
T
F
F
-P
F
T
F
T
-Q
(p ∧
T
F
T
T
P/\-Q
F
F
T
T
T
q) =⇒ ( ~ p ∨ q)
F
T
T
(~P\/Q)
ii. Is the statement in (i) a tautology? Justify your answer.
no because the statement is not all true.
iii. Is the statement in (i) logically equivalent to ∼ p ∨ q? Justify your answer.
yes because the true false statements came out equal.
6. (10 points) Rewrite the following statements and their negations using quantifiers and symbols.
Then, argue whether the statement is true by proving the statement or by finding a
counterexample.
i. Each integer has the property that its square is less than or equal to its cube.
THE NEGATION STATEMENT WOULD PROVE TO BE FALSE
WHEN ATTEMPTED SO
EACH INTEGER HAS THE PROPERTY AND ITS SQUARE IS LESS
THEN ITS CUBE.
ii. There is a rational number which is greater than its square root.
THERE IS A RATIONAL NUMBER AND IT IS NOT GREATER THAN ITS
SQUARE ROOT.
7. (10 points) Consider the following conditional statement:
If n is an odd integer, then n2 > n.
i. Write the converse of the original statement:
IF N^2>N THEN N IS AN ODD INTEGER
ii. Write the negation of the original statement:
n is a odd integer then n^2<N
iii. Write the contrapositive of the original statement:
IF N IS NOT AN ODD INTEGER THEN N^2 IS NOT MORE THEN N
iv. Of (i)-(iii), which is logically equivalent to the original statement?
THE LOGICAL EQUIVALANT WOULD BE THE CONVERSE
v. Provide a value for n that makes the negation of the original statement (your answer for (ii))
true and justify your answer.
1
N(N-1)=0
N=0,1
8. (10 points) Give examples of the following, and provide justification as to why your answers
are examples of the following.
i. Sets C and D such that |C × D| = 12.
12 ELEMENTS =
CXD= { (2,7),(2,8),(2,9),(2,10),(3,7),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10)}
C {(2,3,4})
D{(7,8,9,10)
ii. Sets X, Y , and Z such that {X, Y, Z} is a partition of the set W = {x ∈ Z | 1 ≤ x ≤ 7}. Hint:
It might help to list out the elements of W first, if you can.
iii. A statement involving the variables p, q, r, and s and any of ∼, ∨, ∧, =⇒ , ⇐⇒ that is
TRUE when p, q, r, and s are ALL FALSE.
iv. Real numbers x and y which disprove the following statement:
x
> 1 ∨ (x · y < 0) .
∀x, y ∈ R,
y
v. A set D such that |P(D)| = 16. (remember that P(D) denotes the power set of D).
For questions 9 and 10, give formal proofs. At minimum, I expect to see the following:
• Use complete sentences throughout your proof.
• Define any variables you use, and explain your arguments thoroughly using definitions where necessary.
• Make sure your proof ultimately satisfies the claim you are trying to show.
• End your proof with a square, or any other appropriate celebratory symbol that you like!
9. (10 points) Disprove the following statement:
∀x, y ∈ R, x + y =
p
x2 + y 2 .
10. (10 points) Prove the following:
If m is an odd number and n is an even number, then m − 3n is odd.
Hint: start with the definition of an even number and the definition of an odd number.
95 - 3(4)=83 TRUE
93-3(12)=57 TRUE
63-3(32)=-33 FALSE
37-3*(12)=1 TRUE
SUPPOSE M IS EVEN THEN N IS ODD
THEN M-3N IS EVEN
WE CAN TRY TO PROVE THIS WITH
12-3(37)
WE CAN ALSO TRY SINCE THE
DEFINITION OF ODD STATES WE
CAN TRY 83 WHICH IS NOT
DIVIDABLE BY TWO NUMBERS.