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Math Help > Sets, Set theory, Number systems > Sets
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.
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|
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
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.