MAT 243 HW 3 SOLUTIONS
(1) (4 pts) Fill in the blank in the statements below:
(a) P (S) is the set containing all subsets of S i.e., P (S) = cfw_X | X S
(b) A B = cfw_x | x A x B
/
(c) A function f : A B is one-to-one if and only if a A b A (f (a) = f (b)
2.2 Problem 7 (a): f (x) = 2x3 + x2 log x is O(x3 ). (b): f (x) = 3x3 + (log x)4 is O(x3 ). (c): f (x) = (x4 + x2 + 1)/(x3 + 1) is O(x). (d ): f (x) = (x4 + 5 log x)/(x4 + 1) is O(1). 2.2 Problem 13: We know that 2n 3n for all n 0,
MAT 243 ONLINE WRITTEN HW 6 SOLUTIONS
(1) (4 pts) Fill in the blank in the statements below:
(a) When making a sequence of choices in counting we are using the Product Rule.
(b) We are using the Sum Rule when choosing among mutually exclusive alternatives
6.1-6.2-6.3-6.4
Quiz 7 due Wednesday April 1st
(1) Given 10 chips each of them of different colors (including red and white). For each of the following
questions explain your answer. You should use factorial, combination, permutation or simple products
wh
MAT 243 WRITTEN HOMEWORK 1
NAME: Solutions
(1) Fill in the blank in the statements below:
(a) Two propositions are logically equivalent if and only if they have the same truth values.
(b) A tautology is a a proposition that is always true.
(c) The negatio
Mat 243 Ionascu Test II Review
SHORT STORY of the TOPICS to be covered on Test II
Section 2.3 Functions: understand the definition of a function, domain, codomain, range; what it mean for
a function to be injective (or one-to-one), surjective (or onto) an
MAT 243 ONLINE WRITTEN HW 5 SOLUTIONS
NAME:
(1) (4 pts) Fill in the blank in the statements below:
(a) In an inductive proof verifying the condition for n = 1 (or the lowest possible value)
is called the base step or base case.
(b)-(c) In the induction st
KUNAL LANJEWAR
Base Case:
Every member of the initial population cfw_0 is even.
Inductive Step:
We want to show that both 3 + 2 and ! are even. Now suppose that
is even. Then, 3 + 2 and
MAT 243 Test 1 Practice solutions
1.
(a)
(b)
(c)
(d)
sufficient
necessary
necessary
sufficient
2.
Consider the statement if x>1 then x2>1
3.
Converse: if x2>1 then x>1
False Counter ex: let x= -3.
Necessary cond
Inverse: if x 1 then x21 False Counter ex:
MAT 243 ONLINE WRITTEN HW 5 SOLUTIONS
NAME:
(1) (4 pts) Fill in the blank in the statements below:
(a) In an inductive proof verifying the condition for n = 1 (or the lowest possible value)
is called the base step or base case.
(b)-(c) In the induction st
Discrete Mathematics
Summer 03
Assignment # 3 : Solutions
Section 3.1
12.The sum of any two odd integers is even.
Given: Two odd numbers a and b.
Prove: a+b is even.
From definition of odd number, a=2n+1 and b=2k+1 for integers n, k.
a+b = 2n+1+2k+1 = 2n
Assignment Unit1 Propositional Logic
1. (1 pt)
Enter T for each true proposition, F for each false proposition
and N for each statement which is not a proposition.
1. What time is it?
2. x+y=y+x for every pair of real numbers x and y.
3. All insects are a
KUNAL LANJEWAR
1.
Let,
P be the set of all people
L(x) be the predicate "X likes ice cream"
K(x) be the predicate "X is a kid"
i.
Every kid loves ice cream Premise
x D, K (x) L (x)
ii.
Joey
REVIEW FOR TEST 1, MAT 243, FALL 2013
The exam covers sections 1.1, 1.3, 1.4, 1.5, 1.6, 1.7, 2.1, 2.2, 2.3, and 2.4.
Test 1, October 2, 2013
Think of this review as a starting point for studying for Exam 1.
You will be asked for denitions on the exam. You
1.6 Problem 5 (a): 2 is an integer greater than 1, so 2 is in the set. (b): 2 is not the square of an integer, so 2 is not in the set.
(c): The set {2, {2} does contain 2 (it's listed as the first element). (e): 2 is not an elemen
Mathematics 243 Discrete Mathematical Structures
Professor Tim Callahan Time: T 3:154:30 in PSA 307. Book: Discrete Mathematics and Its Applications, by Kenneth H. Rosen, 5th ed. You should read the relevant section of the book before I go over the s
Assignment Unit2 Propositional Equivalence
1. (1 pt) Complete the following truth table by filling in the
blanks with T or F as appropriate.
p q p!q :p :q :q!:p
TT
TF
FT
FF
p!q and :q!:p are
_ A. not logically comparable
_ B. not logically equivalent
_ C.
Solutions to exam 1
1. Construct the truth table for the following proposition:
(p q ) (p r)
Solution:
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
p q p r p r (p q ) (p r)
T
FF
T
T
T
FT
F
F
F
FF
T
T
F
FT
F
T
T
TF
F
F
T
TT
T
T
T
TF
F
F
T
TT
T
T
2
MAT 243 ONLINE WRITTEN HW 7 SOLUTIONS
(1) (4 pts) Fill in the blank in the statements below:
(a) The general form of the Inclusion-exclusion formula says that |A1 A2 . An | =
n
|Ai |
=
i=1
|Ai Aj Ak | . + (1)n+1 |A1 A2 . An |
|Ai Aj | +
1i<jn
1i<j<kn
(b)
Edward Beickman
Beickman BC Calculus 2012-2013
WeBWorK assignment number Set Theory Problems is due : 07/08/2013 at 02:43pm MST.
Go to Mr. Beickmans Home Page ( http:/ww2.chandler.k12.az.us/Domain/4854) for the course syllabus, grading policy and
other in
MAT 243 Fall 2016
Review for Test 1
1. Suppose A = cfw_a, b, c and B = cfw_b, cfw_c, cfw_a, c. True or false.
(a) B A
(b) B
(c) cfw_b, cfw_c A B
(d) cfw_b, cfw_c B
(e) cfw_c B
(f) |A B| = 5
(g) |A B| = 3
(h) cfw_c, cfw_a, c B A
2. Suppose A = N and B = cf
MAT 243, HW 2, Jile Shi
1.6
14 b)
Let r(x) ber is one of the ve roommates listed,
Let d(x) bex has taken a course in discretemathematics,
Let a(x) bex can take a course in algorithms.
Y: represents an arbitrary person.
x(r(x)d(x)
Hypothesis
r(y) d(y)
1Uni
MAT 243, HW 2, Shi, Jile
1.4
12. abdf
19. a) P(1) P(2) P(3) P(4)P(5)
c) (P(1)P(2)P(3)P(4)P(5)
e)(P(1)P(2)P(4)P(5)( P(1) P(2) P(3) P(4) P(5)
24. Let S(x) = x is a student in your class
C(x) = x has a cellular phone
F(x) = x has seen a foreign movie.
P(x) =
MAT243 HW6
Jile Shi
3.2:
3:
X4+9x3+4x+74x4 for all x>9; witnesses C=4, k=9
8:
a) Since logx grows more slowly than x, x2 logx grows more slowly than x3, so the rst term
dominates. Therefore this function is O(x3) but not O(xn) for any n < 3. 2x3 + x2 logx
Juan Espino
Kolossa MAT 243 ONLINE A Spring 2017
Assignment Unit1 Propositional Logic due 01/12/2017 at 11:59pm MST
1. (1 point)
Enter T for each true proposition, F for each false proposition and N for each statement which is not a proposition.
1.
2.
3.
KUNAL LANJEWAR
1. There are 25 prime numbers in the [1,100] interval.
Let total number of prime number = n.
Therefore n = 25.
Probability of n is not prime =
10025 75 3
=
= =0.75
100
100 4
2. Let,
T = Total number of people = 100
C = Cat owners = 52
D = D
KUNAL LANJEWAR
Base Case:
Every member of the initial population cfw_0 is even.
Inductive Step:
We want to show that both 3 n+2 and n2 are even. Now suppose that
n is even. Then, 3 + 2 and n2 are also even, by definition of even
numbers. Therefore, by the
Anthony John Arellano Practice Proof 1 (Proctored)
Let n be any integer. Then n2+1 < n2 + 2 < n2 + 3.
If k = n2+2, then n2+1 < k < n2+3.
Since n is an integer, n2+2 is also an integer, and k is thus an integer, validating the proof that for every integer
Anthony John Arellano Exam 1
Let n be any positive integer.
Suppose x = n2 and y = n2+3.
Then there is an odd integer z, between x and y
Z=2w+1 for some integer w, so x < 2w+1 < y
Let k be an even integer
2w is even, so n2 < k+1 < n2 + 3
Base Case: 1 divided by 4 leaves a remainder of 1.
n = aq+r
1 = 4*a+1, there exists a so that this is true (a=0), thus base case is true
Structural Induction: By definition, if n= 4a+1, there exists a so that 3n+2 = 4a + 1, and n2 = 4a+1. Now
we have to t
AJ Arellano Written Homework Week #1
1. E
5.
2. A
3. C
4. TFFTF
a. If I live until the age of 120, then I dont smoke
b. If there is a rainbow, then there must be rain
c. If I am able to see the stars, then the sky is clear
d. If we dont develop nuclear po
AJ Arellano Written Homework Week #1
1. E
5.
2. A
3. C
4. TFFTF
a. If I live until the age of 120, then I dont smoke
b. If there is a rainbow, then there must be rain
c. If I am able to see the stars, then the sky is clear
d. If we dont develop nuclear po