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FUNCTIONS:
RELATIONS:
If A and B are two non-empty sets and R is a relation from A to B, then R is a function
if it relates each element of A to a unique element of B.
Let A = {2, 3, 4} and B = {4, 6, 8, 10}, then a few examples of relations from A to B are:
Let A = {2, 3, 4} and B = {4, 6, 8, 10}, then a few examples of relations from A to B are:
- {(x, y)
A x B, y = x + 6}
= {(2, 8), (4, 10)} - {(x, y)
A x B, y ≥ x2}
= {(2, 4), (2, 6), (2, 8), (2, 10), (3, 10)}
TYPES OF RELATIONS:
- Void Relation (Empty Relation):
A relation R in a set A is called Void relation or Empty relation, if no element of set A is related to any element of A.
Hence R = ɸ which is a subset of A x A.
For example:
Let A = {1, 2, 5, 8}, then
R = {(x, y), x, y
A, x – y = 7} is an example of void relation in A, since no ordered pair (x, y)
A x A satisfies x – y = 8.
- Identity Relation:
For a given set A, I = {(a, a), a
A} is called the Identity relation in A. In identity relation every
element of A is related to itself only.
For example:
If A = {2, 3 ,4} then I = {(2, 2), (3, 3), (4, 4)} is the identity relation in A.
- Symmetric Relation:
A relation R in a set A is said to be Symmetric if and only if aRb L bRa for all a, b
A.
For example:
The relation "is equal to" is a symmetric relation for if a = b then b = a.
- Transitive Relation:
A relation in a set A is called Transitive if and only if, (a, b)
R and (b, c)
R then (a, c)
R for all a, b, c
A.
For example:
The relation "is greater than" is a transitive relation, if a > b and b > c, then a > c.
- Reflexive Relation:
A relation R in a set A is said to be Reflexive if and only if aRa, for all a
A. It means, every element of A is related to itself.
For example:
Let A = {3, 4, 5} and R = {(a, a) : a
A}
Hence, R = {(3, 3), (4, 4), (5, 5)} which is a reflexive relation.
- Equivalence Relation:
A relation R in a set A is called an Equivalence relation if and only if
- it is Reflexive that is, (a, a)
R for all a
A.
- it is Symmetric that is, (a, b)
R L (b, a)
R for all a, b
A.
- it is Transitive that is, (a, b)
R and (b, c)
R L (a, c)
R for all a, b, c
A.
For example:
Let T be set of triangles in a plane. Then two triangles t1, t2
T are equivalent if they are congruent (which means they can be put on each other).
Similarly the relations "is similar to" and "is parallel to" are also examples of equivalence relations. - it is Reflexive that is, (a, a)
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