Day 2, Mat420: Real Analysis

I distinguish between them, however: I think of axioms as more primitive. Our author says that Axioms are "statements that cannot be demonstrated in terms of simpler concepts."

We want a set of axioms that is

1. consistent,
2. independent, and
3. few in number, but sufficient in number to lead to an interesting body of results.

Euclid's Postulates for Geometry illustrate this. First, define "postulate" as synonymous with "axiom"!

1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

In terms of independence, many people (including Euclid) believed that the fifth postulate (the "parallel postulate") should be derivable from the others, but in the end, it wasn't! It is independent of the others. Certainly the postulates for geometry are "few in number" but still provide an interesting body of results.

We tend to have few axioms and a plethora of definitions. Our author later defines "collection" and "aggregate" by saying that they are synonymous with set.

Our author notes that definitions are "iff" (if and only if): A circle is a set of point in the plane equidistant from a given point. That is,

A circle set of points in plane equidistant from a given point.

In propositional logic every statement has a truth value. In fact, that's a definition: A statement is a sentence that is unequivically true or false.

Theorems are conditional: given this, we conclude that.

If this, then that.

If A, then C. A implies C.

For theorems, the only interesting case assumes A is a true statement: If [all of this is true], then [that must be true, also].

• The truth table for implication.

• Truth tables for other operations: and, or, negation

• Note that these are functions (mostly bi-variate), on a finite and hence discrete domain.

1. (direct)
2. (contraposition)
3. (contraction)
4. By cases
5. If , and, in addition, the converse is true (), then we say that A and C are equivalent.
6. Finally, the anti-proof technique: counterexample

• Set
  • is taken as an undefined term;
  • contains elements -- except when it doesn't!
  • is well-defined, in the sense that we can always decide if a given object is an element or not.

• Union
• Intersection
• Complement
• Set-difference