# Relation on a Set

Consider two sets A and B. Every subset of the product set A Ã— B is called a relation between A and B. It is denoted by R.

Hence, we can say that .

If (x, y) âˆˆ R, then we can write it as x R y and is read as â€˜x is related to yâ€™.

If A = B, then R is a relation on A.

The set of first elements of the ordered pair that belongs to R is known as the domain of the relation R. It is denoted by D.

D = {x: (xyâˆˆ R, for x âˆˆ Ai.e.

The set of all second elements of the ordered pair that belongs to R is known as the range of the relation R. It is denoted by E.

E = {y: (xyâˆˆ R, for y âˆˆ Bi.e.

Example
Let R be a relation on the set of whole numbers W, defined by a + 4b = 24. Find
(i) R
(ii) Domain of R
(iii) Range of R
Solution
We have R = {(ab) : a âˆˆ Wb âˆˆ Wa + 4b = 24} = {(24, 0), (20, 1), (16, 2), (12, 3), (8, 4), (4, 5), (0, 6)}
The above set of ordered pairs is obtained by taking b = 0, 1, 2, 3, 4, 5, 6 in the given relation and finding the value of a. If b > 6, then the value of â€˜aâ€™ obtained becomes negative which does not belong to the set W.
Domain of R = {24, 20, 16, 12, 8, 4, 0}
Range of R = {0, 1, 2, 3, 4, 5, 6}

# Different Types of Relation

• Identity relation: A relation R on a set is said to be an identity relation, if both the elements of the ordered pairs are same.

The identity relation on a set A is shown as IA = {(xx) : x âˆˆ A}

Example: Let A = {1, 2, 3}, then IA = {(1, 1), (2, 2), (3, 3)}

• Universal relation: A Ã— A is called the universal relation on a set A.

Example: Let A = {2, 4, 6}, then R = A Ã— A = {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)}

• Âƒ Void relation: If a relation R from A to B is a null set, then R is said to be a void relation.

Example: Let A = {1, 3} and B = {9, 13}.

Let R = {(m, n) : m âˆˆ An âˆˆ Bm Ã— n is even}.

Since none of the numbers (1 Ã— 9), (1 Ã— 13), (3 Ã— 9), (3 Ã— 13) is even, R is an empty set.

• Âƒ Reflexive relation: The relation R on a set â€˜Aâ€™ is said to be a reflexive relation, if R contains all possible ordered pairs of the type (xx) for all x âˆˆ A.

Example: Let A = {3, 6, 9}

The relation R = {(3, 3), (3, 6), (6, 6), (9, 9), (9, 6)} is a reflexive relation, since all possible pairs of (xx) type are present in R.

The relation R = {(3, 3), (3, 6), (6, 9), (9, 9), (9, 6)} is a not reflexive relation, since (6, 6) is missing in R.

Note: R is a reflexive relation, if every element in a set â€˜Aâ€™ is related to itself.

• Symmetric relation: The relation R on a set A is said to be symmetric, if (xyâˆˆ R and yxâˆˆ R.

It means that, for each ordered pair (xy) if the reverse pair ( yx) is also there in R, then R will be symmetric.

Example: Let S be a set of all straight lines in a plane. Then the relation R in S defined by, â€˜a is perpendicular to bâ€™. Then the relation R is said to be symmetric, since a straight line â€˜aâ€™ is perpendicular to another line â€˜bâ€™, and the line â€˜bâ€™ is also perpendicular to line â€˜aâ€™.

• Âƒ Transitive relation: The relation R on a set A is said to be transitive if (xyâˆˆ R and ( yzâˆˆ R â‡’ ( xzâˆˆ R

Example: Let there be a set of straight lines in a plane. Then the relation R on the set of lines defined by â€˜a is parallel to bâ€™ is a transitive relation, since if a line a is parallel to a line b and the line b is parallel to another line c, then the line a is parallel to c.

• The relation â€˜a is perpendicular to bâ€™ is not a transitive relation. This is because if a line a is perpendicular to a line b and if the line b is perpendicular to a line c, then a and c will be parallel to each other.
• Equivalence relation: The relation R on a set A is said to be an equivalence relation, if R is reflexive, symmetric and transitive.

Example: The relation R = {(xy) : x âˆˆ Ay âˆˆ Ax = y} is an equivalence relation.

We can see that,

x = xi.e., reflexive

x = y â‡’ y = xi.e., symmetric

x = yy = z â‡’ x = zi.e., transitive

• Âƒ Inverse relation: If R is a relation from the set A to B, then the inverse relation R-1 from B to A is defined by R-1 = {(ba) : (abâˆˆ R}

Example: Let A = {1, 2}, B = {a, b} and R = {(1, a), (2, a), (1, b), (2, b)} be a relation from A to B.

Then inverse relation of R is R -1 = {(a, 1), (a, 2), (b, 1), (b, 2)}.

# Functions

Let A and B be two non-empty sets. Then, if each element x of A is related with a unique element f (x) of B, it is called function from A to B and it is denoted by f A â†’ B.

The element f (x) is called the image of x, while x is called the pre-image of f (x).

Let f A â†’ B, then A is called the domain of f, while B is called the co-domain of f.

The set f (A) = { f (x) : x âˆˆ A} is called the range of f.

Example : Let A = {1, 2, 3, 4}, B = {4, 9, 16, 25}
Let f (x) = x2
Then, f is not a function, since no element of B is assigned to the element 1 âˆˆ A.

Example: Let A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8, 10, 12, 14}
Let f(x) = 2x
Then f(1) = 2
f(2) = 4
f(3) = 6
f(4) = 8
f(5) = 10
We can see that each element of A has a unique image in B.
So f : A â†’ B is a function from A to B.

# Various Types of Functions

• Oneâ€“one function: If distinct elements in A have distinct images in Bi.e.f is â€˜one-oneâ€™ if f(x1) = f(x2â‡’ x1 = x2. It is also called as injective function.

Example: Let fA â†’ B be defined by f(x) = x2.

We know that, 1 â‰  -1, but f(1) = f(-1). So â€˜f â€™ is not oneâ€“one.

Let fA â†’ B be defined by f(x) = x3.

Here, f is oneâ€“one as f(x1) = f(x2â‡’ x1 = x2.

• ÂƒManyâ€“one function: If two or more than two elements of set A have the same image in B, then it is called â€˜manyâ€“oneâ€™ function.

Example: Let fA â†’ B be defined by f(x) = x2.

We know f(-1) = (-1)2 = 1 and also f(1) = (1)2 = 1.

So two elements of set A have the same image in set B. Hence â€˜fâ€™ is a manyâ€”one function.

• Onto function: A function f : A â†’ B is called onto or surjective if f(A) = B, i.e., for all b âˆˆ B there at least one a âˆˆ A with f(a) = b

It is also called as surjective function. In order words, no element in B should be without a pre-image. For onto functions, range = co-domain.

Example: Let A = {1, 2, 3}, B = {a, b} and let f = {(1, a), (2, a), (3, b)}.

Here, the domain is A and no two components of f have the same first element. Hence it is a function. Also to every element y in B, there is an element in A and hence f is an onto function.

• Into function: If at least one element in B has no pre-image in A, it is called into function.

Example: Let A = {1, 2, 3}, B = {a, b, c, d } and let f = {(1, a), (2, b), (3, c)}

It is clear that f is a function from A to B.
Here the element d in B does not have a pre-image in A.

Hence f is an into function.

• Bijective function: A oneâ€“one and onto function is said to be bijective function.
• Equal functions: If two functions f and g have the same domain and satisfy the condition f(x) = g(x), then f and g are said to be equal functions.
• Composite function: Let f A â†’ B and gB â†’ C be two functions. The function from A to C which maps an element x âˆˆ A into gf (x)) âˆˆ C is called a composite function of â€˜f â€™ and â€˜gâ€™. It is represented as â€˜g o f â€™.

Example: Let A = {1, 2, 3}, B = {w, x, y, z}, C = {2, 4, 6, 8, 10}

âˆ´ f = {(1, w), (2, x), (3, y)} and g = {(w, 2), (x, 4), (y, 6), (z, 8)}

Then gof is the function {(1, 2), (2, 4), (3, 6)}

• Inverse function: Let f be a bijective function from A to B and f (x) = y.

Then we can define a function f -1 such that f -1y) = x.

The above function f -1y) is called the inverse of f.

Note: A function has an inverse function if and only if it is bijective.

Example
If f(x) = 2 + x, then f -1(x) = ?
Solution
Let y = f(x)