Lecture 1 Notes Winter 2009 Lecture 1 notes
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MATH 135 Winter 2009
Lecture I Notes
Two of the main purposes of MATH 135 are to teach you about proof (what it means, how to prove
things, etc.) and to teach you about precision in mathematics.
What is a Proof?
A proof is
“A rigorous mathematical argument which unequivocally demonstrates the truth of a
In other words, in mathematics a proof is a sequence of steps, starting with one or more hypotheses,
which follow logically from each other, using the axioms of mathematics, leading to the desired con-
Why Proof ?
We want to know if a statement is always true.
Mathematics is cumulative – we use old results to prove new results. If it turned out that one result
that we thought was true actually failed sometimes, the results following from it could also be false.
What kinds of professions rely on mathematics?
Actually, just about every profession relies on math somehow!
Would you want to drive on a bridge designed by an engineer who relied on mathematics that
she was mostly sure of (rather than on mathematics had been proven)?
Would you trust an actuary to calculate how much you need to contribute to your pension based on
a formula that worked some of the time?
This is why proof is very important from a practical point of view. Proof is also crucial from
an intellectual point of view.
A related notion is that precision in mathematics is vital. It is important that we all speak the
same mathematical language (that is, we understand terms to mean the same thing), that we deﬁne
what needs deﬁning as precisely as possible, and that we say what needs to be said as precisely as
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