A set must be well defined; i.e., for any given object, it must be unambiguous whether
or not the object is an element of the set. For example, if a set contains all the chairs in
a designated room, then any chair can be determined either to be in or not in the set.
If there were no chairs in the room, the set would be called the empty, or null, set; i.e.
one containing no elements. A set is usually designated by a capital letter. If A is the
set of even numbers between 1 and 9, then A = {2, 4, 6, 8}. The braces, {}, are com
monly used to enclose the listed elements of a set. The elements of a set may be de
scribed without actually being listed. If B is the set of real numbers that are solutions
of the equation x2 = 9, then the set can be written as B = {x : x2 = 9} or B = {x | x2 = 9},
both of which are read: B is the set of all x such that x2 = 9; hence B is the set {3, -3}.
Membership in a set is indicated by the symbol ∈ and non-membership by ∉; thus, x ∈
A means that element x is a member of the set A (read simply as “x is a member of A”)
and y ∉ A means y is not a member of A. The symbols ⊂ and ⊃ are used to indicate that
one set A is contained within or contains another set B; A ⊂ B means that A is contained
within, or is a subset of, B; and A ⊃ B means that A contains, or is a superset of, B.
There are three basic set operations:
intersection;
union;
complementation.
The intersection of two sets is the set containing the elements common to the two sets
and is denoted by the symbol ∩. The union of two sets is the set containing all elements
belonging to either one of the sets or to both, denoted by the symbol ∪. Thus, if C =
{1, 2, 3, 4} and D = {3, 4, 5}, then C ∩ D = {3, 4} and C ∪ D = {1, 2, 3, 4, 5}. These
two operations each obey the associative law [any binary operation, symbolized by ∘,
joining mathematical entities A, B and C, obeys the associative law if (A ∘ B) ∘ C = A
∘ (B ∘ C) for all possible choices of A, B and C] and the commutative law [any binary
operation, symbolized by ∘, joining mathematical entities A and B, obeys the commu
tative law if A ∘ B = B ∘ A for all possible choices of A and B], and together they obey
the distributive law [in mathematics, given any two operations, symbolized by * and ∘,
the first operation, *, is distributive over the second, ∘, if a * (b ∘ c) = (a * b) ∘ (a *
c) for all possible choices of a, b and c].
test 1: mathEmatics
In any discussion, the set of all elements under consideration must be specified, and it
is called the universal set. If the universal set is U = {1, 2, 3, 4, 5} and A = {1, 2, 3},
then the complement of A (written A´) is the set of all elements in the universal set
that are not in A, or A´ = {4, 5}. The intersection of a set and its complement is the
empty set (denoted by ∅), or A ∩ A´ = ∅; the union of a set and its complement is
the universal set, or A ∪ A´ = U.