7/28/2019 Basic Set Theory Terminology
1/2
Basic Set Theory Terminology
Sets: A set is a collection of well-defined objects called the
elements or members of theset. The defining property of a set is
that sets having the same elements without regardto order or
repetition are all identical. Thus, {a, b, c} = {b, c, a} = {b, a,
c, a}.
When dealing with large finite or infinite sets, the set can be
identified by a definingproperty. For example, {x | x is a student}
would name the set of all things x such that xis a student. {x |
=df The set of all x with the property
Set membership: a A, read as a is a member of set A or a is an
element of set A. Itis possible for a set to contain other sets as
its members. This statement can be true orfalse depending upon
whether a is an element in set A.
Universal set, V: The set that contains everything in the domain
or the universe ofdiscourse. V =df {x | x = x}
Null (empty) set, : The set that contains nothing. =df {x | x
x}. This set isrepresented by the symbol or by displaying the set =
{ }. {} The null set isa member of every set.
Unit set: Any set that contains exactly one member. {a} =df {x |
x = a}
Unordered pairs: {a, b} Any set that contains exactly two
members.
Ordered pairs: A pair with a specific term order. , {a, b} = {b,
a}
Ordered n-tuples: An ordered set that contains n members.
Inclusion: AB, read A is included in B or A is a subset of B if
and only if everymember of A is also a member of B. {a, b} {a, b,
c, d}. If we assume that AB is true,then (aA)(xB) is true.
Proper Subset: AB, read A is a proper subset of B if and only if
AB but not BA. Inother words, A is a proper subset of B when A is
included in B but B is not included in A.
Power set: (A) This set includes all of the subsets of set A.
(A) = {B | BA} Where Ais {a, b, c}, (A) = { , {a}, {b}, {c}, {a,
b}, {a, c}, {b, c}, {a, b, c} }
Union: AB, read the union of sets A and B is the set that
contains all of the membersof A and all of the members of B. AB =df
{x | xA or xB}. {a, b} {b, c} = {a, b, c}.
If x AB, then (xA) or (xB).Intersection: AB, read the
intersection of sets A and B is the set that contains anymembers
that belong to both A and B. AB =df {x | xA and xB}. {a, b}{b, c} =
{b}
