WEEK
2
M: Addition to Definition 1.19 of primes: A prime is always >
1. Types of proofs: Direct Proof, Indirect proof,
Proof by contradiction. Examples.
1.21. Definition.
Rational number.
1.22. Proposition.
Any rational number can be expressed as a fraction p/q such that
- p is an integer
- q is a positive
integer
- p,q are relative
primes.
(Will be proved
later.)
Example 1.
Given integers x,y: The product x · y is odd <=>
both x,y are odd. (Direct and indirect proof)
Example 2. Square
root of 2 is irrational. (Finish next time.)
PROBLEMS FOR DISCUSSIONS:
Book, p.30 Exercise 57, 58, 59, 60, 62
W: Completion of Example 2. More examples of proof by contradiction.
Example 3. There
are no primes a,b,c such that a3 + b3 = c3.
Fact A:
x1 · x2 · ···
· xn is odd iff all numers x1,x2,
..., xn are odd
Fact B:
x + y is odd iff one of the two numbers x,y is even and the other
odd.
Example 4. If
a,b,c are odd integers then the equation ax2 + bx + c = 0 has
no rational solution.
PROBLEMS FOR DISCUSSIONS:
Book, p.30 Exercise 67, 71, 72, 73 and p.33 Exercise 81, 83,
84
F: Starting Section 2: SETS. Basic notions and notation.
Extensionality and separation.
Example 5. If x1,x2,
..., xn> 1 are integers then the
number x1 · x2 · ···
· xn + 1 is not divisible by any of them.
Fact. Every integer
larger than 1 is divisible by a prime.
Application (Euclid):
There are infinitely many primes.
2.1. Nebulous definition.
Set, membership.
2.2. Basic properties of sets.
(a) Extensionality (b) Separation.
Notation: The empty
set. Standard sets.
PROBLEMS FOR DISCUSSIONS:
Book, p.33 Exercise 88, 89; p.34 Exercise 91 and
p.39 Exercise
92, 94, 99, 100, 102