Sets, Relation & Function

Sets, Relation & Function

Set:

A set is a collection of well defined objects i.e. the objects follow a given rule or rules. If we say that we have a collection of short students in a class, then this collection is not a set as ''short students'' is not well defined. If, however, we say that we have a collection of students whose height is less then 5 feet, then it represents a set.

Elements of a set:

The members of a set are called its elements. A set is usually denoted by capital letters A, B, C etc, where as the elements of a set are generally denoted by lower case letters a, b, c, d etc. If an element x is in set A, we say that x belongs to A and write x A. If the element x is not in A then we write x A.

Examples of sets:

1. The set of vowels in the alphabet of English language.

2. The set of all points on a particular line.

3. The set of all lines in a particular plane.

4. The set of all odd natural numbers.

5. The set of all real numbers.

The elements in a set can be written in any order e.g.

A = {1, 3, 5, 7, 9, ..}, B = {11, 9, 5, 7}, C = {.., - 2, - 1, 0, 1, 2, ..}, D = {Amar, Aman, Ajay}.

This is called the roster method of representing a set. A set can also be represented by stating the properties within braces, which are satisfied by the elements of the set e.g.

A = {x : x = 2n + 1, n 1, n N}, A = {x : 6 x 12, x N}.

This method of representing a set is called the set builder method.


Some special sets:

(i) Finite and infinite sets:

A set A is finite if it contains only a finite number of elements; we can find the exact number of elements in the set. Otherwise, the set is said to be an infinite set


(ii) Null set:

A set which does not contain any element is called a null set and is denoted by. A null set is also called an empty set.


(iii) Singleton set;

A set which contains only one element is called a singleton set.


(iv) Equal sets:

Two set are said to be equal, if every element of one set is in the other set and vice-versa e.g., are equal sets, since the order of the elements is

immaterial.


(v) Equivalent sets:

Two sets A and B are equivalent if the elements of A can be paired with the elements of B, so that to each element of A there corresponds exactly one element of B e.g. ,

are equivalent sets.

Note: Equal sets are equivalent, but equivalent sets may not be equal.

(vi) Subsets:

If each element of a set A is also an element of a set B, then A is called a subset of B, and we write ,

Note: that.

(vii) Proper subsets:

A set A is called a proper subset of B if and only if each element of A is an element of B and there is at least one element of B which is not in A i.e. and and we write

Note: The null set is a subset of every set and every set is a subset of itself.

(viii) Power set:

The power set of a set A is the set of all of its subsets, and is denoted by

Note : The null set and set A are always elements of.

Theorem: If a finite set has n elements, then the power set of A has elements.


Operations on sets:

The operations on sets, by which sets can be combined to produce new sets, can be best illustrated through Venn diagram as shown in side wise figure:




(i) Union of sets:

The union of two set A and B is defined as the set of all elements which are either in A or in B or in both. The union of two sets is written as as shown in side wise figure:



This definition can be extended to the union of more than two set. We define, is this case, the union as

.

Note : (i.e. union of sets is idempotent), and Also and.


(ii) Intersection of sets:

The intersection of two sets A and B is defined as the set of those elements which are in both A and B and is written as



Note : and.

The commutative, associative and distributive laws hold for intersection of two sets i.e.









The intersection of n sets is written as

.



Disjoint sets:

Two set A and B are said to be disjoint, if there is no element which is in both A and B, i.e. e.g. are disjoint.


iii) Difference of sets:

The difference of two set A and B, taken in this order, is defined as the set of all those elements of A which are not in B and is denoted by A - B i.e. =.

Similarly set B - A is the set of all those element of B which are not in A i.e.





iv) Complement of a set:

Complement of a set A is defined as S - A where S is the universal set and is denoted by or i.e. or .



Note: 1., A A^c = S.



2. .






Venn diagram for.



(v) Application:

Let A be a finite set. The number of elements in A is denoted by. Let A and B be two finite sets. If A and B are two disjoint sets, then .

If A and B are not disjoint, then

(i)

(ii)

(iii)

(iv)







(vi) Cartesian product of sets:

Let a be an arbitrary element of a given set A i.e. and b be an arbitrary element of B i.e.. Then the pair is an ordered pair. Obviously . The cartesian

product of two sets A and B is defined as the set of ordered pairs. The cartesian product is denoted by

.

In general and if A or B is a null set then.

Moreover,.

Note : (i)

(ii)

(iii)

(iv)

(v)

(vi)


Relations

Let A and B be two sets. A relation R from the set A to set B is a subset of the cartesian product . Further, if , then we say that x is R-related to y and write this relation

as x R y. Hence .


Domain and Range of a relation: Let R be a relation defined from a A set to a set B, i.e.. Then the set of all first elements of the ordered pairs in R is called the domain of

R. The set of all second elements of the ordered pairs in R is called the range of R. That is,

D = domain of or ,

= range of or .

Clearly and.


Inverse relation:

Let R be a relation from a set A to a set B. Then, the inverse relation of R, denoted by, is a relation defined by ,


(i) Identity relation:

A relation R in the set A defined by or is called the identity relation.

(ii) Void relation:

A relation R in the set A is void relation if.

(iii) Universal relation:

A relation R in the set A defined as is called the universal relation in the set.

(iv) Reflexive relation:

A relation R, in a set A, is called a reflexive relation if for all or for all.

(v) Symmetric relation:

A relation R, in a set A, is called a symmetric relation if or. A relation R in a set A is symmetric if.

(vi) Transitive relation:

A relation R, in a set A, is called a transitive relation if and or and .

(vii) Equivalence relation:

A relation R, in a set A is an equivalence relation if R is reflexive, symmetric and transitive i.e. for all, , and.


Functions

Let A and B be two non-empty sets. Let to each element of A, there corresponds exactly one element of B. This correspondence between the elements of A and B is called a function

from A to B. Function is a special case of a relation. A function from A to B is usually denoted by the symbols f, g etc. and we write. We also say that ''f is a mapping from A

to B''.

The set A is called the domain of the function f and B is called the co domain of the function f.




(i) Graph of a function:

Letbe a function. Then the subset of is called the graph of the function f. We know that a relation R from the set A to B is a subset of . The

relation R is a function from the set A to the set B, if every element of A is the first element of exactly one ordered pair of R. The function f from the set A to the set B is usually written
explicitly.




(ii) Equal functions:

Let f and g be two functions defined from A to B. Then are equal if.

If the function f and g are equal, then the subsets, graph of f and graph of g, of are equal.


(iii) One-to-one functions:

Let be a function from the set A to the set B. Then f is said to be one-to-one function if the images of distinct elements of A are distinct elements of B i.e. if

.



A one-to-one function is also called an injective function.

A function which is not one-one is called a many-one function.


(iv) Onto functions:

Let be a function from the set A to set B. Then, f is said to be an onto function (onto mapping) if every element of B is image of at least one element of A i.e. for, these

exist at least one such that. In other word, the range of f = B.




(v) One-to-one and onto functions:

Let be a function from the set A to the set B. f is said to be one-to-one and onto if it is both one-to-one and onto.

A one-to-one and onto function is also called a bijective function.




(vi) Constant function:

Let be a function from. f is said to be a constant function if there exists an element such that, for all.




(vii) Inverse image and inverse function:

Let be a function from the set A to the set B. Let. The inverse image of the element is the set of all elements of A whose image under the mapping f is b i.e.

. If is one-to-one and onto, then the inverse image is called the inverse function of f.

Since f is onto, for is non-empty. In fact is a singleton set. Hence, for every, there exists a unique element such that.

This correspondence between the elements of B and A is called the inverse function of f and is denoted by. Hence if and only if i.e. is defined only when f

is a bijective function. The function is also a bijective function.


(viii) Identity function:

Let A be a non-empty set. If the mapping is such that each element of the set A is mapped onto itself, then f is said to be an identity function. The identity function is a

bijective function.


(ix) Composite Functions:

Let and be two functions. Let. Then, there exists exactly one image. Also B is the domain of g. Since is a function, this element

is mapped to under the mapping g i.e.. This correspondence between the elements of A and C is called the composite function of f and g and is denoted by

gof i.e. the composite mapping is defined by: such that for all.


Note : The range of f is the domain of g. A is the domain of gof and C is its range.

In general, composite function of two functions is not commutative i.e..

In particular if f is a bijection of A onto itself then,

f^-1 of = f of^-1 = I, where I is the identity function.


Binary Operations:

Let A be a non-empty set. Then a function is called a unitary operation on A. A function is called a binary operation on A. The binary operation is usually

denoted by ''o'' or ''*''. The image of (a, b) (A × A) under the binary operation* is denoted by a* b. Similarly, a function is called an n-ary

operation on A.


Note : Since the addition of two odd integers is even, addition on A (set of all odd integers) is not a binary operation.


Laws of binary compositions:

Let A be a non-empty set and '*' be a binary operation defined on A.

Commutative composition. The binary operation '*' is said to be commutative if a*b = b*a for a, b A.

Associative composition. The binary operation '*' is said to be associative if (a*b) *c = a* (b*c) for a, b, c A.

Identity element. An element e A is said to be an identity element for the binary operation if a*e = a = e*a for a A. For binary operation of addition in R, 0 (zero) is the identity

element. For multiplication, the identity element is 1. For a binary operation, if the identity element exists, then it is unique.