Linear Algebra

A-6
The other thing is that we sometimes use induction to go down, say, from
10 to 9 to 8, etc., down to 0. This is OK—‘next number’ could mean ‘next
lowest number’. Of course, at the end we have not shown the fact for all natural
numbers, only for those less than or equal to 10.
Contradiction. Another technique of proof is to show something is true by
showing it can’t be false.
The classic example is Euclid’s, that there are infinitely many primes.
Suppose there are only finitely many primes p1,...,pk. Consider p1 ·
p2 ...pk +1. None of the primes on this supposedly exhaustive list divides
that number evenly, each leaves a remainder of 1. But every number is
a product of primes so this can’t be. Thus there cannot be only finitely
many primes.
Every proof by contradiction has the same form—assume the proposition is
false and derive some contradiction to known facts.
Another example is this proof that √ 2 is not a rational number.
Suppose √ 2=m/n. Then
2n 2 = m 2 .
Factor out the 2’s: n =2 kn · ˆn and m =2 km · ˆm. Rewrite:
2 · (2 kn · ˆn) 2 =(2 km · ˆm) 2 .
The Prime Factorization Theorem says there must be the same number
of factors of 2 on both sides, but there are an odd number (1 + 2kn) on
the left and an even number (2km) on the right. That’s a contradiction
so a rational with a square of 2 cannot be.
Both these examples aimed to prove something doesn’t exist. A negative
proposition often suggests a proof by contradiction.
Sets, Functions, and Relations
Sets. The perfect squares less than 20, the roots of x 5 − 3x 3 + 2, the primes—
all are collections. Mathematicians work with sets, collections that satisfy the
Principle of Extensionality stated below.
A set can be given as a listing between curly braces: {1, 4, 9, 16}, or, if that’s
unwieldy, by using set-builder notation: {x � � x 5 − 3x 3 +2=0} (read “the set
of all x such that ... ”). We name sets with capital roman letters: P =
{2, 3, 5, 7, 11,...} except for the set of real numbers, written R, and the set
of complex numbers, written C. To denote that something is an element (or
member) of a set we use ‘ ∈ ’, so 7 ∈{3, 5, 7} while 8 �∈ {3, 5, 7}.