INTRODUCTION TO PROOF
This is Razvan Gelca's webpage dedicated to teaching Introduction to
Proof
during the Coronavirus pandemic.
Before you proceed, please read the syllabus,
which can be found here.
- Here are the photographs of the white board for the lecture on
01/21:
one, two,
three, four,
five, six.
- Here are the photographs of the white board for the lecture on
01/26:
one, two,
three, four,
five, six.
- Here are the photographs of the white board for the lecture on
01/28:
one, two,
three, four,
five, six.
- Here are the photographs of the white board for the lecture on
02/02:
The first image shows a direct proof in the context of functions:
image one.
The next two images show direct proofs of a special type, proof by example:
image two. Example 2 solves
the problem by guessing two values of \(x\) that yield the same value
of \(y\). The solution to Example 3 contains two parts, one in which
you investigate (the so called heuristics), in one in which you
display the result. Basically, you ask yourself what are the numbers
that make the function fail to be onto, and then you display one of
those numbers to prove that the function is not onto.
The same idea repeats in Problem 4, you first explain why all values of
the function are greater than or equal to -13/4, then choose the number
-10, which is less than -13/4, and conclude that -10 is not a value of \(f\): image three.
Finally the last three images show how to do direct proofs by dividing
into cases. The division into cases by using remainders is motivated by the
statement of the problem: if you have to show that an integer is divisibly
by 6, in our case \(n(n+1)(n+2)\), then better work with remainders obtained
when dividing by 6. image four,
image five, image six.
- Here are the photographs of the white board for the lecture on
02/04:
one, two,
three, four,
five.
- Notes for 02/09:
one, two,
three, four,
five, six, seven.
- Notes for 02/11:
one, two,
three, four,
five, six.
- Notes for 02/16:
one, two,
three, four,
five, six, seven, eight, nine.
- Notes for 02/18:
one, two,
three, four,
five, six.
- Notes for 02/23:
one, two,
three, four,
five,six.
- Notes for 02/25:
one, two,
three, four.
- Notes for 03/02:
one, two,
three, four,
five,six, and also the pdf file. For
the best experience, download this file, open it in full screen mode, and advance
page by page.
- Notes for 03/04:
one, two,
three, four,
five.
- Notes for 03/09:
one, two,
three, four,
five.
- Notes for 03/11:
one, two,
three, four,
five, six.
- Notes for 03/16:
one, two,
three.
- Notes for 03/18:
one, two,
three, four,
five, six.
- Notes for 03/23:
one, two,
three, four,
five.
- Notes for 03/25:
one, two,
three, four,
five.
- Notes for 03/30:
one, two,
three, four,
five.
- Notes for 04/01:
one, two,
three, four,
five, six.
- Notes for 04/06:
one, two,
three, four,
five.
- Notes for 04/8:
one, two,
three, four,
five, six.
- Notes for 04/13:
one, two,
three, four,
five, six.
- Notes for 04/15:
one, two,
three, four,
five, six.
- Notes for 04/22:
one, two,
three, four,
five, six.
- Notes for 04/27
one, two,
three, four,
five.
- Notes for 04/29:
one, two,
three, four,
five.
Homeworks
The final exam is here and is due
on Friday, May 07, at 6pm.
Practice problems
