5.4 Strong Mathematical Induction
Principle of Strong Mathematical Induction
Let P (n) be a predicate that is defined for integers n, and let a and b be fixed integers with
a b. Suppose that the following two statements are true:
Step 1 (Basis Step): P (a
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:
mcquain@math.vt.edu
Office Hours: To be announced
Tutoring Paid tutoring in the math de
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 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 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
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
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 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 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 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 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 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 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 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
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
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
Math 2534
Solution to Homework 9 on Sets
Problem 1: Use elements to prove the following theorems:
Theorem 1: If A C and B D, A D C B for all sets A, B, C and D
Proof:
x A D x A x D by definition of difference
x C x D since A C
x C x B since B D
x C B
Math 2534 Test 2B solution Fall 2013
Problem 1: Prove the following using PMI and clearly indicate where you use the
inductive hypothesis and use definitions appropriately.
Theorem: If given that a1 4, a 2 12, and a n a n-1 a n-2 , n 3, then a n is always
Chapter 3. The Logic of Quantified Statements
3.1 Predicates and Quantified Statements I
Predicate calculus is the symbolic analysis of predicates and quantified statements. Consider the
sentence
Daniel is a graduate of Virginia Tech.
Daniel is the subjec
Chapter 4. Number Theory & Methods of Proof
4.1 Direct Proof and Counterexample
We begin by reviewing some definitions about integers.
Definitions
The integers are
under addition, subtraction, and multiplication. That is to say,
given any two integers m a
2.3 Valid and Invalid Arguments
In mathematics and logic we define an argument as a series of statements followed by a
conclusion.
Definition
Argument: A sequence of statements consisting two parts: Premises and a Conclusion
Argument Form: A sequence of s
4.4 Direct Proof and Counterexample IV:
Division into cases and The Quotient-Remainder Theorem:
When an integer n is divided by any positive integer d, the result is a quotient q and a
nonnegative remainder r that is smaller than d.
The Quotient-Remainder
Chapter 2. The Logic of Compound Statements
2.1 Logical Form and Logical Equivalence
Symbols:
Set of Numbers
Irrational Numbers
Natural Numbers
Symbol
Description
Symbol
Whole Numbers
/
Integers
Rational Numbers
Irrational Numbers
Meaning
Symbol
Descripti
Chapter 7. Functions
7.1 Functions Defined on General Sets
Definitions:
from a set X to a set Y , denoted f : X Y , is a relation from
, to Y , the
, that satisfies two properties:
A
X, the
1. every element in X is related to some element in Y
2. no elem
6.3 Disproofs, Algebraic Proofs
Previously we have only seen statements that are true. 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
2.2 Conditional Statements
1. Conditional statement
Logical inference is subject to a grammatical flow.
We reason from a
to a
. Logically we represent this as
connective
p
hypothesis
q
conclusion
Definition
A conditional statement is if p, then q or p imp
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 person who loves all fruits.
How does the arrangement of the q
Math 2534 Homework 5
Sequences and PMI
Show all work! STAPLE multiple sheets!
Problem 1: Proof by contradiction. Use definitions only
Theorem: For all natural numbers n, 5n + 3 is not divisible by 5.
Proof: Assume that there exist at least one natural num
Math 2534 Solutions Homework 1 Fall 2016
Problem 1: Use Truth Tables to verify the following: Be very detailed and explain
your conclusion.
1) (p q) p q
p
q
T
T
T
F
F T
F F
p q
F
F
F
T
p q ( p q) p q
T
F
F
F
T
T
Continue to look at all possible truth val
Math 2534 Solution to Homework 8 on Sets
Spring 2013
Problem 1: Given sets A cfw_a, b,cfw_c, c, B cfw_a,cfw_b, c, b, , C cfw_b, c,cfw_
Find the following: (dont forget to use equal signs.)
a) Find the following sets:
1) A B cfw_a, b
2) B C cfw_b, c
3) A B
Math 2534 Solutions to Homework 3 on Quantifiers
Problem 1:
Put the following sentences into symbolic logic using single quantifiers. Define your variables,
the domain and the predicate.
Part A:
a) Discrete math students love math.
Let D be the set of all
Math 2534 Solution Homework 8 on Sets
Fall 2014
Problem 1: Given sets A cfw_a, b,cfw_c, c, B cfw_a,cfw_b, c, d , , C cfw_b, c
Find the following: (dont forget to use equal signs.)
a) Find the following sets:
1) A B cfw_a
2) B C cfw_a, b, c,cfw_b, c, d ,
Math 2534 Sol Homework 4 Methods of proof
Instructions: Prove or give a counterexample. Use your own paper and present a well written
argument for each theorem proved. Justify all assertions made. Use the direct method unless
told otherwise.
Problem 1:
Th