Sets & Domains

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Sets & Domains are just fancy ways of grouping numbers together to be used for something later. Sometimes, we want to limit the number of choices that can be used for solving problems. Inequalities do basically the same thing, but include larger groups of numbers (real numbers, rational and irrational numbers, etc...)

1 Sets and elements

Sets and elements

A mathematical set is defined as an unordered collection of distinct elements. That is, elements of a set can be listed in any order and elements occurring more than once are equivalent to occurring only once.

We say that an element is a member of a set. An element of a set can be anything. It's easiest to begin with only numbers as elements. For that reason, most of the examples in this book will only include numbers, but this is only a technique to make the topic less abstract.

Terminology

For a set A having an element x, the following are all used synonymously:

x is a member of A

x is contained in A

x is included in A

x is an element of the set A

A contains x

A includes x

Notation

We specify a set by specifying its members. The curly brace notation is used for this.

$\{1,\ 2,\ 3\}\,\!$

is the set containing 1, 2, and 3 as members. Or, {mother, this ipod, my school, the planet Jupiter, 12} is also a set. The curly brace notation can be extended to specify a set by specifying a rule for set membership.

$\{x\ |\ x = 1\ \mathrm{or}\ x = 2\ \mathrm{or}\ x = 3\}\,\!$

is again the set containing 1, 2, and 3 as members.

$\{x\ |\ x\ \mathrm{is\ a\ natural\ number}\}\,\!$

is a set of all natural numbers. This form or representing set can be generalized as:

$\! \{x\ |\ P(x)\}$

P(x) is a statement about the variable x. The set defined by above notation is a set of all objects, x can be substituted with such that P(x) is true.

A modified epsilon notation is used for set membership. Thus

$x \in A\,\!$

says that x is a member of A. We can also say that x is not a member of A:

$x \notin A\,\!$

Characteristics of sets

A set is uniquely identified by its members.

$\{x\ |\ x\ \mathrm{is\ an\ even\ prime}\}\,\!$

$\{x\ |\ x\ \mathrm{is\ a\ positive\ square\ root\ of}\ 4\}\,\!$

$\{2\}\,\!$

Moreover, the sets A and B are said to be equal if and only if every element of A is also an element of B, and every element of B is an element of A.

All the above expressions specify the same set even though the concept of an even prime is different from the concept of a positive square root. Repetition of members is inconsequential in specifying a set. The expressions

$\{1,\ 2,\ 3\}\,\!$

$\{1,\ 1,\ 1,\ 1,\ 2,\ 3\}\,\!$

$\{x\ |\ \mathrm{x\ is\ an\ even\ prime\ or}\ x\ \mathrm{is\ a\ positive\ square\ root\ of}\ 4\ \mathrm{or}\ x = 1\ \mathrm{or}\ x = 2\ \mathrm{or}\ x = 3\}\,\!$