MATH 135
Fall 2008
Lecture II Notes
Patterns and Conjectures
We often use patterns in mathematics to guess what is happening in general (that is, make a “con
jecture”), and then try to prove our pattern/conjecture (that is, turn it into a theorem).
Consider the following problem:
n
points are chosen on the circumference of a circle in such a way that when all pairs of
points are connected, no three of these lines intersect at a single point. How many regions
are formed inside the circle?
Types of Statements to Prove
There are two basic types of statements that we try to prove in mathematics:
1) Simple statements:
A
eg. “
√
2 is irrational”
2) Conditional statements:
A
⇒
B
(“
A
implies
B
” or “If
A
then
B
”)
eg. “If
n
is an odd integer, then
n
2

1 is divisible by 4.”
Here,
A
= “
n
is an odd integer” is our hypothesis and
B
= “
n
2

1 is divisible by 4” is our
conclusion.
A
and
B
are both statements on their own about some unknown number
n
.
To prove propositions of Type 1, we use the axioms of mathematics, logic, our ingenuity, and other
results we know to be true.
To prove propositions of Type 2, we use the hypothesis (or hypotheses) and the list of tools above.
In doing so, we assume that
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 Fall '08
 ANDREWCHILDS
 Math, Algebra, Trigonometry, Parity, Evenness of zero

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