MAT 137Y – An Introduction to Proofs
One of the challenges that most students encounter in this course is the ability (or lack thereof) to
write proofs. This is because most high schools tend to focus on the computational aspect of mathematics.
However, university mathematics also deal with analysis and theory; we formulate mathematical questions
and use logic to provide rigourous solutions or ideas.
It is likely that this is your ﬁrst encounter with theorems and proofs, and as such, we recommend that
you read this material carefully. You should pay particular attention to the actual theorems and proofs that
are illustrated in this handout. Sometimes you may have to read a proof more than once to understand it,
which is normal. More importantly, articulating proofs is a skill that you will need to master if you are to be
successful in this course. You will often be asked to demonstrate these skills in your problem sets.
We will assume that everyone is familiar with the different sets of numbers: the real numbers
R
, the
rationals
Q
, the irrationals, the natural numbers
N
, and the integers
Z
. These are deﬁned in Section 1.2 of
Salas, Hille, and Etgen. We will also assume you have some familiarity with the mathematical notation that
you have learned in high school. For example,
Q
=
{
x
∈
R

x
=
p
q
,
p
,
q
∈
Z
;
q
6
=
0
}
,
which says: the rationals is the set of all real numbers
x
such that
x
can be written in the form
p
/
q
, where
p
and
q
are integers, and
q
is not zero. The irrationals are deﬁned as
Q
=
{
x
∈
R

x
6∈
Q
}
,
that is, the set of all real numbers
x
such that
x
is not rational. One question which should spring to mind is:
how do we know there are any irrational numbers at all? Later we will prove that
√
2 is one such number
that is irrational.
We will now focus on the basics of mathematical statements, theorems, and proofs. If you have not done
so already, please read Section 1.8 of the textbook. In a nutshell, consider the statement
If it rains, then the baseball game is cancelled.
The statement consists of a hypothesis (“It rains”) and the conclusion (“The baseball game is cancelled”).
Whether the statement is true depends on the truth of the hypothesis and conclusion. The only case where
the statement is false is when the hypothesis is true and the conclusion is false; in other words, the statement
is false if it rains and the baseball game is not cancelled.
To simplify things a bit, we often use the notation
A
⇒
B
to denote “If
A
, then
B
.” The
converse
of the
statement
A
⇒
B
is deﬁned to be the statement
B
⇒
A
. A statement and its’ converse are not necessarily
equivalent. For example, if the baseball game is cancelled, it does not mean that it is raining. If a statement
and its’ converse are equivalent, we use the notation “
A
⇔
B
” to denote “
A
if and only if
B
” (“
A
⇒
B
” and
“
B
⇒
A
”). Finally, the
contrapositive
of the statement
A
⇒
B
is the statement “not