Math 2534 Solutions Test 1B Fall (green test)
Problem 1:(18pts) Use Algebra of Logic to simplify the following and justify each step.
Theorem: [ ( p q) q] p q p
Proof:
[ ( p q) q] p
Given
[ ( p q) q]
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 be
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
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
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
2.3 Valid and Invalid Arguments
Math 2534
Order of Operations
Order of Operations
Order
Operations
1.
Always evaluate negation first
2.
Second, evaluate and , when both are present
you should use pa
Math 2534
2.2 Conditional Statements
Example 3: Are the statements
p _ q and p ! q equivalent?
p
T
T
F
F
q
T
F
T
F
Important
If p then q is logically equivalent to
Example 4: Write in the form If
dism
2.2 Conditional Statements
Math 2534
Digression
Sets of Numbers and Notation
Symbol Meaning
2
Useful symbols
(can use them in your Hw and quizzes)
)
*
Set Name
Symbol
Natural Numbers:
N
Whole Numbers:
Math 2534
5.1-5.2. Sequences, Explicit Formulas and Recursive Definition
Explicit Formulas for Sequences
Example 1: Here are two examples of sequences:
3
(i) ,
4
4
,
5
5
,
6
.
(ii) 1,
1
,
2
1
,
4
1
,
Math 2534
4.2 Direct Proof and Counterexample II
Rational Numbers
Recall the following
Definitions
A real number r is
if, and only if, it can be expressed as a quotient
of two integers (with a non ze
Math 2534
4.3 Direct Proof and Counterexample III
Divisibility
Definitions
Consider the
n is
n and d, with
if, and only if,
NOTATION: d | n. We have that d | n
, then we say that
Other ways to say n i
Math 2534
3.3 Statements with multiple quantifiers
Statements with multiple quantifiers
Example 1: Consider the statement:
For each student in this section of Math 2534, there is a homework problem t
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
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 quant
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)
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 o
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 i
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 equiv
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 n
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
functio
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)
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 fo
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
a
Math 2534
3.2 Negations of Quantified Statements
Negation Universal Statement and Existential Statements
Example 1: What is the negation of the statement below?
All the students in this class are wear
4.4 Direct Proof and Counterexample IV
Math 2534
The Quotient-Remainder Theorem
Quotient-Remainder Theorem
integers
Given any integer n and a positive integer d,
such that
n = dq +
and
r<
The proof wi
MATH 2534
Homework Assignment 5
Due: Mon, February 27, 2017
Section: 4.1
1. Prove or disprove the following:
(a) For some integer n > 2, n3 8 is prime.
(b) For all integers n and m, if n + 3m is even,
3.3 Statements with Multiple Quantifiers
Previously we have considered statements with a single quantifier. We now consider
statements, which are a little more complex. For example,
There is a differe
MATH 2534
Homework Assignment 2
Due: Mon, February 6, 2017
Section: 2.3
1. Determine whether the following argument is valid or invalid using a truth table. Justify your answer.
q rp
pr
qp
pq
2. Consi
Math 2534
3.4 Arguments with Quantified Statements
Arguments with Quantified Statements
We are going to discuss argumentations for statements with quantifiers.
The key principle to do this is called u
Math 2534
2.1 Logical Forms and Logical Equivalence
A large part of this course consists of developing your ability to think abstractly. Well
learn how to deal with symbolic representations and use di