Most, if not all, of
pure mathematics is couched in the language of sets. You may notice that this
section contains many definitions and only a few theorems. However, even a
definition can contain a lot of mathematical wisdom. It took mathematicians
centuries to formulate some fundamental definitions.
Set:
A set is
a collection of items considered as a whole. If there are only a few items, the
set can be defined by listing them in braces. For example, the set A might
be defined as follows:
A = {1, 2, 3}
Two sets are equal if
they contain exactly the same elements. That is, set A is
equal to set B if every element of A is also
an element of B, and every element of B is also an
element of A. The order in which the elements of a set are listed
in its definition is irrelevant. For example, the sets {1, 2, 3} and {3,
2, 1} are equal.
An element cannot
belong to a set more than once. Therefore, when a set is defined by listing its
elements, each element is listed only once.
A set that contains
no elements is called the empty set, and is represented by the
symbol ∅.
If every element of
the set A is also an element of the set B,
then A is said to be a subset of B,
represented symbolically by A⊆ B, or B is
said to include A. Every set is a subset of itself, and
the empty set is a subset of every set.
If A⊆ B and there is at least one element of B that is
not an element of A, then A is said to be a proper subset
of B, represented symbolically by A⊂ B.
A subset is often
defined by some property of its elements. For example, let A = {1, 2, 3,
4, 5, 6}, and let B = {2, 4, 6}. Then B could
be defined as the set of all elements of A which are even, or in
symbols:
B= {x∈ A | x is even}.
Here the symbol |
means "such that". The word "all" is understood. In some
cases the set A may also be understood.
The intersection of
any number of sets is the set of elements that they all have in common. For
example, the intersection of {1, 2, 3, 4, 5}, {2, 3, 4, 5, 6, 7,
8, 9} and {3, 5, 7, 9} is {3, 5}. It is clear
that the intersection of a collection of sets is a subset of every set in the
collection. The intersection of two sets A and B is
represented symbolically by A∩B.
The intersection
operation has several obvious properties:
·
Commutativity: A∩B = B∩A.
·
Associativity: (A∩B)∩C =
A∩(B∩C).
·
A∩B = A if, and only if, A⊆ B.
The union of
any number of sets is the set of all of their elements. For example the union
of {1, 2, 3, 4, 5}, {2, 3, 4, 5, 6, 7, 8, 9} and {3,
5, 7, 9} is {1, 2, 3, 4, 5, 6, 7, 8, 9}. It is clear that every
set in a union is a subset of their union. The union of two sets A and B is
represented symbolically by A∪ B.
The union operation
has several obvious properties:
·
Commutativity: A∪ B = B∪A.
·
Associativity: (A∪ B)∪ C = A∪(B∪C).
·
A∪B = B if, and only
if, A⊆ B.
Two sets are said to
be disjoint if they have no elements in common; i.e., A and B are
disjoint if A∩B = ∅. Three or more sets are said to be disjoint if
every two of them are disjoint.
The notation A-B is
used to indicate the set of all elements of A that are not
elements of B. This operation has no standard name, but when B is
a subset of A, A-B is sometimes said to be the complement
of B in A.
The relationships
among sets are often represented pictorially by a Venn diagram, in
which sets are represented as the interiors of overlapping circles (or other
plane figures). Set combinations are represented by areas bounded by the
circles, as shown in the following example for two sets:-
Ordered Pairs
An ordered
pair is a set of two elements in a specified order. An ordered pair is
usually written (a,b) where a is the first
element and b is the second element. Two ordered pairs (a,b) and (c,d) are
equal if a=c and b=d. Reversing the elements of an
ordered pair produces a different ordered pair if the elements are not the
same. For example, the ordered pair (1,2) is not equal to the
ordered pair (2,1).
For two sets A and B,
the cross product A⨯ B is the set of all ordered pairs whose first
and second elements are elements of A and B,
respectively. That is,
A⨯ B = {(a, b) ∣ a∈ A and b∈ B}
Ordered triples,
quadruples, etc. could be defined, but they are seldom needed.
Relations
A relation R on
a set A is simply a set of ordered pairs of elements of A,
i.e., R ⊆ A⨯ A. Two elements a and b are said to
obey the relation if (a,b) is in R. However, for
most relations, the set notation is not used. Instead, a symbol such as ~ placed
between the elements to indicate that they obey the relation; for example a~b means
that (a,b) is in R.
Other symbols often
used for relations are
= > < ≥ ≤ ∣ ≠ ⊃ ⊂ ⊇ ⊆ ≡
Most useful relations
have some additional properties. A relation ~ on the set A is
equivalence if the following hold for every a, b and c in A:
·
It is reflexive: a~a.
·
It is symmetric: a~b implies
that b~a.
·
It is transitive: a~b and b~c imply
that a~c.
A set of nonempty
subsets of a set A is called a partition of A if
each element of A belongs to one and only one of the subsets;
i.e., if the subsets are disjoint and their union is A. The
following theorem establishes a connection between an equivalence relation and
a partition.
Theorem: If ~ is an equivalence relation on the
set A, then there is partition of the set A such that a~b if, and only if, a
and b belong to the same set in the partition. Conversely, if P is a partition
of A, then "a and b belong to the same set in P" is an equivalence
relation.
Proof. Consider the set P of subsets Ta =
{x ∈ A | x~a}. Clearly every a in A belongs
to at least one subset in P, namely Ta.
Hence the sets in P are nonempty and their union is A.
Now let Ta and Tb be
two subsets in P. If they have an element c in
common, then c~a, c~b and x~b for
every x ∈ Tb. By transitivity x~a and x
∈ Ta, too. Similar arguments show that every element of Ta is
also an element of Tb. Hence Ta and Tb are
equal. If two subsets in P have no element in common, they are
disjoint. Hence P is the desired partition.
The converse is
trivial.
The sets in the
partition associated in this way with an equivalence relation are called
its equivalence classes. They are often used to define mathematical
systems.
Equivalence relations
on two sets A and B can be used to define an
equivalence relation on A⨯ B in the obvious way: (a,b) is
equivalent to (c,d) if a is equivalent
to cand b is equivalent to d.
Order
A partial
order on a set A is a relation ≤ with the following
properties for every a, b and c in A:
·
It is reflexive: a
≤ a.
·
It is antisymmetric: a
≤ b and b ≤ a imply that a = b.
·
It is transitive: a ≤
b and b ≤ c imply that a ≤ c.
A partial order ≤ on
the set A is called a linear order (or
a total order) if, for every two elements a and b of A, a
≤ b or b ≤ a (or both, if a = b).
The set of all
subsets of a set is partially ordered by inclusion: S ≤ T means S⊆ T. This partial order is usually not a total order because we can find
two subsets, such as {1,2,3} and {2,3,4}, such
that neither is a subset of the other.
The familiar relation
≤ in arithmetic is a total order.
In working with a
partial or total order, it is common to define some associated relations:
·
a ≥ b means b
≤ a,
·
a < b means a ≤ b and a ≠ b,
·
a > b means b ≤ a and b ≠ a.
There is an
alternative way to define partial and total orders. A relation < is a
partial order if obeys the following two conditions:
·
It is transitive: a < b and b
< c implies that a < c.
·
a < a is always false.
A partial order is a
total order if it is also trichotomous: for any two elements a and
b, one and only one of the following holds:
·
a < b,
·
a = b,
·
b < a.
The other relations
are then defined in terms of <:
·
a ≤ b means a
< b or a = b.
·
a ≥ b means b
< a or a = b.
·
a > b means b < a.
It can be shown that
the two ways of defining partial and total orders are equivalent.
Generally, the names
"partial order" and "total order" are applied to the entire
set of relations ≤, <, > and ≥ without
specifying which is the order relation and which are associated with it.
Operations
A unary
operation on a set is a function whose domain is that set. What
distinguishes a unary operation from an ordinary function is the notation used,
and often its relationships with other functions or operations. For example,
the function that carries any real number x to the
number -x is a unary operation called negation.
The range of the function is often the same set, but this is not required.
A binary
operation is a function whose domain is the cross product of two sets
(or the cross product of a set with itself). For example, addition and
multiplication are two binary operations on the set R⨯ R, where R is
the set of real numbers. The image of an ordered pair (x,y) is
usually written as x+y for addition and xy for
multiplication. Here x and y are called
the operands. The former notation is usually used only for
addition, or operations very much like addition. The latter notation is used
for more general operations.
Unary and binary
operations are very common in mathematics; operations with three or more
operands are rare, except for extensions of binary operations as noted below.
Binary operations on a single set are more common than binary operations on
pairs of sets, but both are encountered frequently.
A binary operation is
said to be associative if it can be used on three operands
without regard to their grouping, i.e., if
(xy)z = x(yz)
(x + y) + z = x + (y + z)
If a binary operation
is associative, we can write the result for three operands without parentheses,
making it a well-defined operation with three operands:
xyz = (xy)z = x(yz)
x + y + z = (x + y) + z = x + (y + z)
Ordinary addition and
multiplication are associative; so are many other binary operations. The
composition of functions is an associative binary operation (provided the
functions have suitable domains and ranges). In fact, most binary operations
are associative.
If a binary operation
is associative, it is easy to extend the property to operations on four or more
operands:
wxyz = (wxy)z = (w(xy))z = w((xy)z) = w(xyz).
A binary operation
is commutative if the order of the operands does not affect
the result; i.e., if
xy = yx
x + y = y + x
If a commutative
operation is also associative, commutativity can easily be extended to
operations on three or more operands:
xyz = (xy)z = (yx)z = yxz = y(xz) = (yx)z = yxz, etc.
Binary operations
which are commutative but not associative are very rare. Binary operations
which are associative but not commutative are fairly common. Composition of
functions is one example; a demonstration of that fact will be given later.
Now consider a binary
operation on A⨯ B with its range in C (which need not be different
sets). Suppose there are equivalence operations on these sets (which also need
not be different), and the binary operation preserves equivalence, i.e.,
the operation when applied to equivalent operands gives equivalent results,
or a ~ b and c ~ d imply that ac ~ bd.
Then, just as a function of one variable was extended to a function on
equivalence classes, an operation on two variables can be extended similarly.
If a and b are any elements of the
equivalence classes P and Q, respectively,
then PQ is defined to be the equivalence class
containing ab. The new operation inherits many properties of the
old one, including associatively and commutatively.
Utility of “set theory” in an organisation
A company consists of sets of resources like personnel,
machines, material stocks and cash reserves. The relationship between these
sets and between the subsets of each set is used to equate assets of one kind
with another. The subsets of highly skilled production workers, within the sets
of all production workers, are critical subset that determines the productivity
or other personnel. Certain subsets of company products are highly profitable
or certain material may be subject to deterioration and must be stocked in
greater quantities than others. Thus the concept of sets is very useful in
business management.
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