COT 3100H
Introduction to Discrete Structures
Exam #1 Solutions
Name: _
1) (10 pts) Here is the truth table for the & operator:
&
False
True
False
True
False
True
True
True
Complete the following truth table to evaluate the boolean expression
(p (q & r) (
COT3100
6)
a) | (A B) (A C)| = | A B | + | A C | - | A B C |
=5+3-2=6
These are the number of items in either sets A and B or sets A and C. Thus, the number
of items ONLY in set A is 8 - 6 = 2.
| (B C) (A B)| = | B C | + | A B | - | A B C |
=6+5-2=9
So, t
COT3100h
3c 2c 5c 129c
24
+
=
6) a) Sum of all the probabilities = c + 2c + +
, thus c =
.
2
3 24
24
129
b) Pr( X 3) =
57c 57
=
24 129
5
d) E ( X ) = x(
x =1
c) Pr( X = 4 | X 3) =
Pr( X = 4) 16
=
P ( X 3) 57
5
cx 2
cx 3
9c 8c 25c
5
317
)=
= c + 4c +
+
= 1
COT3100H, Spring 2014
Assigned: 03/21/2014
Due: 03/26 in class
Assignment #7a
Instructions: Write your answer neatly and concisely. All proofs need to be justified step by step by
using the appropriate definitions, theorems, and logical reasoning. Illegib
COT3100H, Spring 2014
Assigned: 02/26/2014
Due: 03/12 in class
Assignment #6
Instructions: Write your answer neatly and concisely. All proofs need to be justified step by step by
using the appropriate definitions, theorems, and logical reasoning. Illegibl
COT3100H, Spring 2014
Assigned: 02/19/2014
Due: 02/26 in class
Assignment #5
Instructions: Write your answer neatly and concisely. All proofs need to be justified step by step by
using the appropriate definitions, theorems, and logical reasoning. Illegibl
COT3100H, Spring 2014
Assigned: 1/10/2014
Due: 1/22 in class
Assignment #1
Instructions: Write your answer neatly and concisely. All proofs need to be justified step by step by
using the appropriate definitions, theorems, and logical reasoning. Illegible
COT3100H, Spring 2014
Assigned: 1/22/2014
Due: 1/29 in class
Assignment #2
Instructions: Write your answer neatly and concisely. All proofs need to be justified step
by step by using the appropriate definitions, theorems, and logical reasoning. Illegible
COT3100H, Spring 2014
Assigned: 1/29/2014
Due: 2/05 in class
Assignment #3
Instructions: Write your answer neatly and concisely. All proofs need to be justified step by step by
using the appropriate definitions, theorems, and logical reasoning. Illegible
COT3100H, Spring 2014
Assigned: 02/12/2014
Due: 02/19 in class
Assignment #4
Instructions: Write your answer neatly and concisely. All proofs need to be justified step by step by
using the appropriate definitions, theorems, and logical reasoning. Illegibl
COT3100
Page 134
2) a) True
e) True
b) True
f) False
c) True
g) True
d) True
h) True
4) a) False
e) True
b) False
f) True
c) True
d) True
12)
12 12!
a) =
6 6!6! , since we are choosing 6 items out of 12.
6 6
b) , since we choose 2 odd integers out of 6
COT3100
24) We must prove two things:
a) if n is even, then 31n+12 is even
b) if 31n+12 is even, then n is even
We prove the (a) as follows:
Since n is even, we can express it as 2a, where a is an integer.
Thus, 31n+12 = 31(2a)+12 = 62a+12 = 2(31a+6). Sin
COT 3100H Introduction to
Discrete Structures
Final Exam
Name : _
1) (10 pts) Find a closed form solution for the following recurrence relation without
using generating functions:
t(0) = 7, t(1) = 18, t(n) = 5t(n-1) 6t(n-2), for all integers n>1.
2) (10 p
CO 3100 Quiz #1:
Solutions
1) Write out a truth table to evaluate the logical expression (p q) r, for boolean
variables p, q and r.
p
F
F
F
F
T
T
T
T
q
F
F
T
T
F
F
T
T
r
F
T
F
T
F
T
F
T
p q
T
T
F
F
T
T
T
T
(p q)
F
F
T
T
F
F
F
F
(p q) r
T
T
F
T
T
T
T
T
2)
COT 3100 Quiz #2 Solutions
1) Prove the following, using the rules of inference:
[ (p (q r) (p s) (t q) s] [r t]
(Note: There are four premises given above.)
1. p s
Premise
2. s
Premise
3. p
Rule of Disjunctive Syllogism (with #1,#2)
4. p (q r)Premise
5.
COT 3100 Quiz #3 Solutions
1) Determine the truth of the following statements. For all statements, provide a counterexample or proof to show why the statement is true or false. Let x and y come from the
universe of real numbers.
a) xy [x + y = 0]
This sta
COT 3100 Quiz #4 Solution
1) Consider an ant moving on the Cartesian plane, starting from location (0,0). The ant
always moves in the direction of the positive x-axis or positive y-axis, one unit, per
move. The ant's final destination is (7, 9) and it is
COT 3100 Quiz #5 Solutions
1) Let A and B be arbitrary sets. Prove or disprove the following proposition:
Power(A B) = Power (A)- Power (B)
This is false. Let A = cfw_1,2 and B = cfw_2, then A-B = cfw_1, Power(A-B) = cfw_ , cfw_1
Power(A) = cfw_ , cfw_1,
COT 3100 Quiz #6 Solutions
1) You have been put in charge of buying chips for a party. You are told to buy 20
individual bags. You can choose from Cheetos, Lays, Baked Lays, Wavy Lays, Doritos,
and 3-D Doritos. You are also told that you must buy one of e
COT 3100 Quiz #7 Solutions
1) A four-sided die A is labeled with the numbers 2, 2, 2, and 6. Another four-sided die B
is labeled with the numbers 3, 3, 5 and 5. Let X be the discrete random variable
representing the sum of the values shown face up upon ro
COT 3100 Quiz #8
Given two 2x2 matrices, multiplication of these matrices is defined as follows:
a b e
c d g
f ae + bg
=
h ce + dg
af + bh
cf + dh
Using this information, make a conjecture about the value of the expression below (ie.
determine a, b, c,
COT3100
Page 54
4) a) r q
b) r p
c) (r s) q
6) a) True
b) False
c) True
8h)
p
F
F
F
F
T
T
T
T
q
F
F
T
T
F
F
T
T
r
F
T
F
T
F
T
F
T
pq
T
T
T
T
F
F
T
T
qr
T
T
F
T
T
T
F
T
pr
T
T
T
T
F
T
F
T
(pq) (qr)
T
T
F
T
F
F
F
T
[(pq) (qr)] (pr)
T
T
T
T
T
T
T
T
Page 66
1
COT3100
8) a) True
b) False: For x=17, q(x) is true but p(x) is false
e) True
f)True
g) True
h) False, For x=-17, p(x)
q(x) is true but r(x) is false
26)
lim r
n
n
c) True
d)True
L > 0k > 0n[ n > k | rn L | ]
Extra problem:
Use proof by contradiction an
COT3100H, Spring 2014
02/10/2014
Exam #1
Instructions: Closed-book, closed-notes, no calculators; a reference sheet containing all relevant
definitions and theorems will be provided on the test. Write your answer neatly and concisely in the
space provided