Linear Algebra

Appendix
Introduction
Mathematics is made of arguments (reasoned discourse that is, not pottery
throwing). This section is a reference to the most used techniques. A reader
having trouble with, say, proof by contradiction, can turn here for an outline of
that method.
But this section gives only a sketch. For more, these are classics: Propositional
Logic by Copi, Induction and Analogy in Mathematics by Pólya, and
Naive Set Theory by Halmos.
Propositions
Thepointatissueinanargumentistheproposition. Mathematicians usually
write the point in full before the proof and label it either Theorem for major
points, Lemma for results chiefly used to prove others, or Corollary for points
that follow immediately from a prior result.
Propositions can be complex, with many subparts. The truth or falsity of
the entire proposition depends both on the truth value of the parts, and on the
words used to assemble the statement from its parts.
Not. For example, where P is a proposition, ‘it is not the case that P ’istrue
provided P is false. Thus ‘n is not prime’ is true only when n is the product of
smaller integers.
We can picture the ‘not’ operation with a Venn diagram:
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Where the box encloses all natural numbers, and inside the circle are the primes,
the dots are numbers satisfying ‘not P ’.
To prove a ‘not P ’ statement holds, show P is false.
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