MT2002 Analysis

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## Some set theory

There are serious and subtle difficulties in dealing with sets, so we will skirt round these.
(See Some Early History of Set Theory.)

For the moment, think of a set as a collection of elements.
We will assume familiarity with some of the following examples.

Examples

The natural numbers N = {1, 2, 3, ... }
The integers Z = { ..., -2, -1, 0, 1, 2, ... }
The rational numbers Q = { a/bR | a, bZ, b ≠ 0}

All these sets lie inside the set of real numbers R which is what this course is about. We will examine this in more detail later.

Some notation

If A, B are sets then AB and AB are the usual union and intersection.

A × B (the Cartesian product) is the set of all ordered pairs: { (a, b) | aA, bB }.

When we talk about the Real number line R, we use the following notation for intervals.

An open interval: (a, b) = { xR | a < x < b} does not contain its end points.

A closed interval: [a, b] = { xR | axb} does contain its end points.

Some other intervals: [a, b), (a, b], (-∞, b], (a, ∞), etc.

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JOC September 2001