Introduction to Abstract Mathematics
MA 103
Solutions to exercises 9
1
(a) The key point is that, if we allow pairs ( a, b) with b = 0, then the relation R is no longer
an equivalence relation, since it wouldnt be transitive.
If necessary, you should have
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 3
1
(a) To prove : for all n N, n3 + 5n is a multiple of 3.
Proof : Let P(n) be the statement that n3 + 5n is a multiple of 3.
Base case: P(1) is true, since 13 + 5 1 = 6 is a multiple of
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 5
1
2
(a) For all n N we get ()(n) = ( (n) = (n2 ) = 2n and ( )(n) = (n) =
(2n ) = (2n )2 = 22n (= 4n ).
(b) Since we have ()(1) = 21 = 2 and ( )(1) = 22 = 4, we have that ()(1) 6= ( )(1),
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 7
1
(a) The statement a|b means: there is a q Z so that b = qa.
The key point for the rest of this question is that you have to engage with this definition,
rather than write down what you
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 2
1
(a) (i)
We can write this set as A = cfw_ x N | x = 5y for some y N .
But there are many other ways; for instance:
A = cfw_ m N | there is a natural number k such that m = 5k ,
or
A =
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 8
1
(a) All these are done by repeatedly finding quotient and remainder using the base number
as divisor. So for base 2 we do
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Introduction to Abstract Mathematics
MA 103
Solutions to exercises 4
1
(a) For every cat born at time n, there are two cats born at time n + 1. So if Cn is the number
of cats born at time n, then Cn+1 = 2Cn . With C1 = 1, this number doubles in each time
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 6
1
First we note that if the person picks 3 socks, he might end up with one blue, one red and
one grey one. So at least 4 socks will be needed.
Below well give two proofs of the fact that
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 1
1
The statement if n is a multiple of 16, then n is not a multiple of 6 , is a statement of the
form if P, then Q. A counterexample is a value of n making P true (where P says n is a
mul
Introduction to Abstract Mathematics
MA 103
2015/16
Solutions to Exercises
This document contains answers to selected algebra exercises from the second half of MA103,
including all the exercises set for classes. There may well be some errors; please do le
BACKGROUND
In this handout, we discuss some points of elementary logic that are apt to cause confusion,
and also introduce ideas of set theory, and establish the basic terminology and notation.
This is not examinable material, but read it carefully, as th
GREEK LETTERS
Learn the names of Greek letters not just the obvious ones like alpha,
beta, and pi, but the more obscure psi, xi, tau, omega. [. . . ] Learn not just
to recognize these letters, but how to pronounce them. Even if you are not
reading mathema
An Approach to Lowercase letters Capital letters
Greek Lettering
Michael A. Covington
Program in Linguistics
The University of Georgia
Copyright 2001.
Noncommercial reproduction for
teaching purposes permitted.
This is a system of Greek
hand-lettering tha
Introduction to Abstract Mathematics
MA 103
Exercises 4
1
Consider female cats that reproduce by the following rule: A cat born at time n has two
daughters both born at time n + 1, and otherwise no further female offspring. Starting with
a single female c
MA103 Slide set 6
Functions and Counting
1
Topics
Functions and their set-theoretic definition
Composition of functions
Surjective, injective, and bijective functions
Inverse functions
Cardinality (= size) of a set
The pigeonhole principle (PP)
Applicatio
MA 103 Introduction to Abstract Mathematics
Extra Examples Session 1
Department of Mathematics
London School of Economics and Political Science
Some practicalities
This term, those sessions are given by me,
Jan van den Heuvel.
Next term it will be the MA1
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 2
1
(a) (i)
We can write this set as A = cfw_ x N | x = 5 y for some y N . But there are
many other ways; for instance :
A = cfw_ m N | there is a natural number k such that m = 5 k ,
or A
MA103 Slide set 2
Statements, Proof, Logic
1
Abstract Mathematics
Mathematics is about making precise mathematical statements
and establishing, by proof or disproof, whether these statements
are true or false.
We start by looking at what this means, conce
MA 103 Introduction to Abstract Mathematics
Extra Examples Session 2
More Examples of Induction
Department of Mathematics
London School of Economics and Political Science
The Principle of Induction
Principle of Induction
Let P (n) be a statement involving
MA103 Slide set 5
Fibonacci Numbers, Using Strong Induction,
Partial Orders
1
Roadmap
Geometric proofs
The Fibonacci numbers and solving linear recurrence equations
Strong induction applied to splitting bread, and chocolate
Partial orders
Strong induction
MA103 Slide set 6
Functions and Counting
1
Topics
Functions and their set-theoretic definition
Composition of functions
Surjective, injective, and bijective functions
Inverse functions
Cardinality (= size) of a set
The pigeonhole principle (PP)
Applicatio
MA103 Slide set 5
Fibonacci Numbers, Using Strong Induction,
Partial Orders
1
Roadmap
Geometric proofs
The Fibonacci numbers and solving linear recurrence equations
Strong induction applied to splitting bread, and chocolate
Partial orders
Strong induction
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 3
1
(a) To prove : for all n N, n3 + 5 n is a multiple of 3.
Proof :
Let P(n) be the statement that n3 + 5 n is a multiple of 3.
Base case: P(1) is True, since 13 + 5 1 = 6 is a multiple o
Example of a relation on N
Consider the following relation
on N :
is even .
For instance,
related by .
and
, but
, that is,
and
are not
This relation has some special properties.
Since
is even for all
(reflexivity)
in N, we have
.
4
Symmetry
We continue t
MA103 Slide set 7
Equivalence Relations
1
Topics
Binary relations
Equivalence relations
Examples of equivalence relations
Equivalence classes
Partitions
Functions define equivalence relations
2
Relations in general
If S is a set, a relation R on S is a su
Introduction to Abstract Mathematics
MA 103
Solutions to exercises 1
1
The statement if n is a multiple of 16, then n is not a multiple of 6 , is a statement of the
form if p then q. A counterexample is a value of n making p true (where p says n is a
mult
1(a) Suppose that n = (2m)2 . Then n = 4m2 = (2m2 1) + (2m2 + 1), showing that it
is a sum of two consecutive odd integers. So S is true.
The converse is: If an integer is the sum of two consecutive odd numbers, then it is the
square of an even integer. T
1.
(a) If n is even, then n = 2k for some integer k, and so n3 = 8k 3 = 2(4k 3 ) is even.
The converse is: If n3 is even, then n is even.
The contrapositive of the converse if: If n is not even, then n3 is not even, That is, it is
If n is odd, then n3 is