# document.write (document.title)

A Set is a collection of objects.  It is an abstract concept, which means the objects are conceptual things, not physical things.  The objects can be anything, even other sets.  This section describes what a set is, defines the notation for describing sets, and gives some examples of different kinds of sets.

### Contents of this section:  Set Description Notation Construction Is 0 a Natural Number? Rational and Irrational ZF Set Theory ZFC: Axiom of Choice

Set Description Notation gives a rundown of the various ways to describe a set -- Roster notation, Set-builder notation, Interval notation, and Graphically.

Set Construction is the construction of the traditional sets: Integers, Reals, Complex, etc.

Topology is an introduction to the basic definitions of terms in the study of metric sets.

The most important notion of set theory is membership.  x is a member of S, x in S, is written with a funny e.  In these notes it is written as x : S.  Two sets S and T are equal just when, for any x, (x is a member of S) is logically equivalent to (x is a member of T). Formally,

S = T means ((x : S <=> x : T), for any x)

S is a subset of T just when, for any x, (x is a member of S) ==> (x is a member of T).

S is a superset of T just when T is a subset of S.

S is a strict subset of T if it is a subset of T and also S ~= T.

Subset is a transitive relation.

Sets can be constructed by giving a property which their elements must satisfy. For example,

m..n = {x : x is a natural number and m <= x <= n}

Standard sets include

N the natural numbers, {1, 2, 3, ...}

Note: some authors include zero as a natural number.

Z the integers {0, 1, -1, 2, -2, ...}

Q the rational numbers

R the real numbers

The empty set {} can also be written as a funny phi, or in this document, /0.

Standard operations on sets include

union A \_/ B = {x : x in A or x in B}

intersection A /^\ B = {x : x in A and x in B}

difference A \ B = {x : x in A and x not in B}

symmetric difference A (+) B = {x : x in A or x in B but not both}

powerset P(A) = {x : x subset of A}

cartesian product A >< B = {(a,b) : a in A and b in B}

The pair (a,b) is an ordered collection of two values. That

is to say (a,b) = (c,d) <=> a = c and b = d.

There is often a universal set U which contains all possible values of interest. It is then possible to define

complement A^c = {x : x not in A} = {x : x in U and x not in A} = U \ A

These operations satisfy the following laws

A \_/ A = A

A \_/ /0 = A

A \_/ U = U

A /^\ A = A

A /^\ /0 = /0

A /^\ U = A

A \_/ B = B \_/ A

A /^\ B = B /^\ A

A \_/ (B \_/ C) = (A \_/ B) \_/ C

A /^\ (B /^\ C) = (A /^\ B) /^\ C

A \_/ (B /^\ C) = (A \_/ B) /^\ (A \_/ C)

A /^\ (B \_/ C) = (A /^\ B) \_/ (A /^\ C)

A^c^c = A

U ^c = /0

/0^c = U

A \_/ A^c = U

A /^\ A^c = /0

(A \_/ B)^c = A^c /^\ B^c

(A /^\ B)^c = A^c \_/ B^c

A \ B = A /^\ B^c

A (+) B = (A /^\ B^c) \_/ (A^c /^\ B)

A subset B <=> A \ B = /0 <=> A /^\ B^c = /0

The size of a set is also known as its cardinality. This is the number of (distinct) members.

cardinality |A| = the number of members of A

Some laws are

|{}| = 0

|{x}| = 1

x ~= y ==> |{x,y}| = 2

(and so on for sets with 3 or more elements)

|A| + |B| = |A \_/ B| + |A /^\ B|

|P(A)| = 2^(|A|)

|A >< B| = |A| * |B|

### Related pages in this website

A "large" set is one whose sum-of-reciprocals diverges; a "small" set is any set that is not large.

Is Zero a Natural Number? -- a discussion of the fact that some authors include zero, and others do not.

The Peano Postulates -- Proving the properties of natural numbers using the Peano Postulates, which have been formulated so that zero is not included in the set of natural numbers.  (There's quite a debate about this point.)

Introduction to Counting -- explains what mathematicians mean by "counting" -- that is, putting sets in one-to-one correspondence.

Construction -- Construction of sets of numbers, starting with the original Peano Axioms, formulated so that zero is included in the set of natural numbers.

Number systems -- construction, definition, description of nonstandard number systems e.g. hyperreals, p-adic numbers

Counting Ordered Pairs of Integers -- An explanation of the "square spiral" that puts the set of natural numbers in one-to-one correspondence with the set of rational numbers.

Abstract Algebra -- definitions of terms associated with Ring Theory and Group Theory

The webmaster and author of this Math Help site is Graeme McRae.