# Calculus

## Set Theory # Introduction ot Sets

## The Concept of a Set

The basic concepts of set theory were created and developed in the late $$19\text{th}$$ century by German mathematician Georg Cantor $$\left( {1845 - 1918} \right).$$

According to Cantor's definition, a set is any collection of well defined objects, called the elements or members of the set.

Sets are usually denoted by capital letters $$\left( {A,B,X,Y, \ldots } \right).$$ The elements of the set are denoted by small letters $$\left( {a,b,x,y, \ldots } \right).$$

If $$X$$ is a set and $$x$$ is an element of $$X,$$ we write $$x \in X.$$ The symbol $$\in$$ was introduced by Italian mathematician Giuseppe Peano $$\left( {1858 - 1932} \right)$$ and is an abbreviation of the Greek word $$\epsilon\sigma\tau\iota$$ - "be".

If $$y$$ is not an element of $$X,$$ we write $$y \notin X.$$

## Defining Sets

There are two basic ways of describing sets - the roster method and set builder notation.

### Roster Method

In roster notation, we just list the elements of a set. The elements are separated by commas and enclosed in curly braces. For example, $$X = \left\{ {1,5,17,286} \right\}$$ or the set of natural numbers including zero $${\mathbb{N}_0} = \left\{ {0,1,2,3,4, \ldots } \right\}.$$

In a listing of the elements of a set, each distinct element is listed only once. The order in which elements are listed does not matter. For example, the following sets are equal:

$\left\{ {5,5,6,7} \right\}\;and\;\left\{ {5,6,7} \right\};$
$\left\{ {1,2,3} \right\},\left\{ {2,3,1} \right\}\;and\;\left\{ {3,2,1} \right\}.$

When writing infinite sets and there is a clear pattern to the elements, an ellipsis (three dots) can be used.

### Set Builder Notation

In set builder notation, we define a set by describing the properties of its elements instead of listing them. This method is especially useful when describing infinite sets.

The notation includes on or more set variables and a rule defining which elements belong to the set and which are not. The rule is often expressed in the form of a predicate. The set variable and rule are separated by a colon ":" or vertical slash "|".

#### Examples:

1. The set of all uppercase letters of the English alphabet.
$U = \left\{ {x\,|\,x \text{ is an uppercase letter of the English alphabet}} \right\}$
2. The set of all prime numbers $$p$$ less than $$1000.$$
$P = \left\{ p\,|\,p \text{ is a prime number and } p \lt 1000 \right\}$
3. The set of all $$x$$ such that $$x$$ is a negative real number.
$X = \left\{ {x\,|\,x \in \mathbb{R} \text{ and } x \lt 0} \right\} \text{ or } X = \left\{ {x \in \mathbb{R}\,|\,x \lt 0} \right\}$
4. The set of all internal points $$\left( {x,y} \right)$$ lying within the circle of radius $$1$$ centered at the origin.
$C = \left\{ {x,y \in \mathbb{R}\,|\,{x^2} + {y^2} \lt 1} \right\}$

## Universal and Empty Sets

A set which contains all the elements under consideration is called the universal set and is denoted as $$U.$$ The universal set is problem specific. For example, if a question is related to numbers, the universal set can be either all natural numbers $$\left( {U = \mathbb{N}} \right),$$ or all integers $$\left( {U = \mathbb{Z}} \right),$$ or all rational numbers $$\left( {U = \mathbb{Q}} \right),$$ etc. - depending on the context.

There is a special name for the set which contains no elements. It is called the empty set and is denoted by the symbol $$\varnothing.$$ There is only one empty set.

## Subsets

A set $$A$$ is a subset of the set $$B$$ if every element of $$A$$ is also an element of the set $$B.$$ A subset is denoted by the symbol $$\subseteq,$$ so we write $$A \subseteq B.$$ If a set $$C$$ is not a subset of $$B,$$ we write: $${C \not\subseteq B.}$$

The empty set $$\varnothing$$ is a subset of every set, including itself.

The sets $$A$$ and $$B$$ are equal if simultaneously $$A \subseteq B$$ and $$B \subseteq A.$$ Notation: $$A = B$$.

A set $$A$$ is called a proper subset of $$B$$ if $$A \subseteq B$$ and $$A \ne B.$$ In this case, we write $$A \subset B.$$ Some authors use the symbol $$\subset$$ to indicate any (proper and improper) subsets.

The power set of any set $$A$$ is the set of all subsets of $$A,$$ including the empty set and $$A$$ itself. It is denoted by $$\mathcal{P}\left( A \right)$$ or $${2^A}.$$ If the set $$A$$ contains $$n$$ elements, then the power set $$\mathcal{P}\left( A \right)$$ has $${2^n}$$ elements.

See solved problems on Page 2.