Math 2534 Solution Homework 1 (sec 2.1) Spring 2015
Problem 1:
Let p, q and r be the propositions given below:
p: The birdwatchers are in the woods.
q: A blue heron was seen at the lake.
r: Migration starts soon.
Part A:
Put the sentences below into symbo
Math 2534 Solutions Homework 6
Put all work on another sheet of paper. Use complete sentences to tie your work together and
make it understandable.
Problem 1: Use PMI to prove that Geometric Series below is valid.
Theorem: 1 a a 2 a3 . a n 1
an 1
, n N
a
Math 2534 Test 1B Spring 2015
Name _
No Electronic Devices. Show and justify all work to get complete credit.
Problem 1: Given the following true statements, convert each statement to symbolic logic
argument form and determine the validity of the argument
Math 2534 Solution Test 2B Spring 2015
Problem 1: (20pts) Use PMI to prove the following theorem and be clear where you use the
inductive assumption. Justify all steps with complete sentences.
Theorem: For all natural numbers n 5, 2n2 n!
Proof: To verify
Math 2534 Spring 2015
Information and Course Policy
CRN: 14660 meets MW at 2:30pm in Torg 3100
Instructor: Margaret McQuain
Office:
McBryde 543
Phone:
231-8277
e-mail:
[email protected]
Office Hours: To be announced
Tutoring Paid tutoring in the math de
Statement of Understanding
for the Expectations in Math 2534
Spring 2015
By my signature below I verify that I have read and understood the policy and
requirements for this class. In particular I understand the honor policy regarding the my
work on the re
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6.1 Set Theory: Denitions and the Element Method of Proof
Denition
A set is a collection of items, and an elemenr is one such item.
We denote such a collection of items with U.
If a is an element of a set S, we write a 6' S and if it is not, we write a 4
7.3 Composition of Functions
Denition
Let f : X > Y and g : Y > Z be functions With the property t at the range of f is a
subset of the domain of 9. Dene a new function g o f : X > Z as follows:
(9 o f)( )= g(f(m) for 811137 E X-
The function g o f is c
6.3 Disproofs, Algebraic Proofs
What do we do when one of these statements is false? Recall that to disprove universal
statements, we prove the negation, which is an existential statement. That is, we nd one
element in our domain for which the statement i
Math 2534 Solution Homework 4 on Proofs
Be precise on domain used and definitions. The universal quantifier must be clearly implied for
valid theorems and the existential quantifier clearly implied for counter-examples. Define your
variables clearly and u
Math 2534 Solution Homework 2 Sec 2.1-2.3
Problem 1: Use Algebra of Logic to Prove the following:
a) Theorem 1:
(p q) ( p r ) (p q) p r
Proof:
( p q) ( p
r ) ( p
[( p q) ( p
[ p (q
[ p] ( p
given
r)
r)
by Commutative and Associative laws
by Distributi
Homework Requirements
1) Use regular sized paper (8.5 X 11)
2) No rough edges on papers torn out of spiral notebooks.
3) Staple multiple papers and put name on all sheets. If a page is lost
then you have a zero for that work.
4) No Ink
5) Work must be nea
Math 2534 Solution Worksheet:
Big O definition:
If f(x) and g(x) are real valued functions then f(x) is Big O of g(x) iff there exist
constants C and K so that
f ( x) C g ( x) , x > K
A) Let f ( x) 2 x2 3x 6 and g ( x) x 2 and verify using the definition
Math 2534 Homework 3
Quantifiers and proofs
Put all work on another sheet of paper and be neat. Staples multiple sheets
Problem 1:
A) Put the following sentences in symbolic logic using a single quantifier.
a) No child will read the dictionary.
Let C be t
Math 2534 Solution Homework 5 on Proof
Follow the outline for proof write up. Justify your steps and use complete sentences.
A note on the method of contradiction:
Contradiction means
(x, P( x) Q( x) x P( x Q( x)
To do a proof of contradiction you start w
Math 2534 Solution Homework 9
Theorem 1: For all sets A, B,
( A B)C AC BC
Proof:
x, x ( A B )C x ( A B )
by the definition of complement
( x A B) by definition of negative
( x A x B) by definition of intersection.
( x A) ( x B) by DeMorgan's Law in log
Math 2534 Solution Homework 7 on PMI Spring 2015
Theorem 1: n N , 1 2 22 . 2n1 2n 1
Proof by PMI: The hypothesis is true for at least one value on n.
Consider the value n = 1 to see that 20 21 1 and it is clear that 1= 1 . In order to see the
pattern clea
Math 2534 Solutions to worksheet on Chap 8
1) Let A = Z Z . Define a relation R on A as follows: for all (a, b) and (c, d) in A,
(a, b) R (c, d) if (Correction) a + d = b + c. Prove that R is an equivalence relation
on A.
Proof: Reflexive: (a, b) Z Z , (a
Math 2534: Lecture Sheet on Methods of Proofs:
Outline for proof process:
1) Always write a clear statement of the conjecture which must explicitly or
implicitly use the universal quantifier.
2) All notation and variables must be clearly defined.
3) (Be s
Math 2534 Solutions Homework 11
Functions and sets Spring 2015
Show all work and staple multiple sheets.
Problem 1;
Let C and D be subsets of A so that A C D and f : C B and g : D B . Define a
function h(x) as follows:
f ( x) if x C
h( x )
g ( x) if x
7.2 One-toOne and Onto, Inverse Functions
Denitions
Let f be a mapping from a set X to a set Y. We say that f is onefo-one or
We if, and only if,
VC17$2 E X, If f($1) = f(1132), then (31 = 1132.
quxzex c; me x men memz) 0P)
What does it mean for a funct
6.4 Boolean Algebras
Denition
A BOOlQOh Q l ge bra (6, *1 X) is a set B together with two
operations, generally denoted + and X, such that for all a, b E B, both
oa+bEB, oaXbEB
and the following properties hold:
1. Commutative Laws: For all a, b E B,
(a)a
7.1 Functions Dened on General Sets
0 omam
Denitions \L (01100110: n
o A 'Funcon from a set X to a set Y, denoted : X > Y is a relation
from X, the domain , to Y, the Co-domaln , that satises two
properties:
[11 every element in X is related to some eleme
Math 2534 Test 2 Solution Summer 2014
Problem 1: (12pts)
Using a direct method prove the following using definitions only (state them correctly.)
Theorem: Let a be an integer. If 2 (a 1), then a is odd.
Proof: Given 2 (a 1) , by definition of divisible th
Math 2534 solution Test 1B Spring 2016
Problem 1: Determine if any of statements below are equivalent. Define your variables. Put the
following statements in implication form, and then justify your conclusion in terms of the
sufficient and necessary condi
Math 2534 Solution Test 1A Fall 2015
(8pts)
Problem 1:
Given the following information, translate the symbolic logic into a unambiguous conversational
English sentence.
Let S be the set of all simple solutions and x is an arbitrary solution.
Let P be the
Math 2534 Solution Test 1 Fall 2016
Problem 1: Determine which of the following statements are equivalent statements by
putting each statement into symbolic notation and define variables and justifying
all steps and conclusion.
Solution: (14pts)
Define A
Math 2534 Test 1 Solutions Spring 2008
Problem 1: (8pts) Given the following statements:
p: Mary is a Math major
q: James is a CS major
r: Karla is an Engineer
w: Harry is a Hokie
Given the statement: (p q) (r w) is false and p is false, determine if r, p
S-17
Math 2534 Test 2 (82pts)
Time Started
7 April 2017
Time Ended
NAME
KEY
Team Name
Use only methods from class. The use of calculators is prohibited on this test. Justify your answers.
Honor Pledge:
I have neither given nor received help on this exam.
F-16
Math 2534 Test 3 (83pts)
Time Started
2 December 2016
Time Ended
NAME
KEY
Team Name
Use only methods from class. The use of calculators is prohibited on this test. Justify your answers.
1. [4pts] Evaluate the maps at the indicated values:
(a) Let A =