We are
related to many peoples in our daily life, and we know what relation in daily
life means so now let us talk about mathematic relations.
Relation and Its Type
Cartesian product:
A mathematical operation that returns a set from multiple sets is called Cartesian product in set theory. That is for sets A and B the Cartesian product AxB is the set of all ordered pairs (a,b)
where a ɛ A and b ɛ B.
For example:
Let us take two sets A and B.
Where A= {1,2,3,4} and
B= {a,b,c,d}
Then the cartesian product of A and B is
denoted by AxB and given by:
AxB=
{(1,a), (1,b),
(1,c), (1,d), (2,a), (2,b), (2,c), (2,d), (3,a), (3,b), (3,c), (3,d), (4,a), (4,b),
(4,c), (4,d)}
Relation:
A Relation R
among sets A and B is a subset of the Cartesian product of the sets A and B.
For example:
R={(1,a), (1,b), (1,c), (2,b), (2,c) , (3,a),
(3,b), (3,c)} is a relation.
A relation R
among sets A and B is a subset of the Cartesian product of the sets A and B.
For example:
R={(1,a), (1,b), (1,c), (2,b), (2,c) , (3,a),
(3,b), (3,c)} is a relation.
Types of Relation:
There are different
types of relation we will explain about them in here:
1. Void or empty relation:
If no element of set A is related to any element of set B, the relation in set A is called Void relation or Empty relation, . Hence R={0} which is
subset of AxB.
2. Identity Relation:
For a given set A , I={(a,a): a
ɛ A} is called Identity relation in A.A is related to itself only, in identity relation.
Example:
If A={ a,b,c,d} then R={ (a,a), (b,b), (c,c), (d,d)} is identity relation
in A.
3. Symmetric Relation:
A relation R in a set A is said to be symmetric relation in A if and only if a is related to b implies b is
related to a.
Example:
The relation “equals to” is symmetric relation as a=b implies b=a.
4. Transitive Relation:
A relation in set A is called Transitive relation if and only if a
related to b and b related to c implies a related to c.
Example:
The relation “greater than” is a Transitive relation as a>b and b>c
implies a>c.
5. Reflexive Relation:
A relation R in a set A is said to be Reflexive if and only if a is
related to a for all a belonging to set A.
Example:
If A={ 1,2,3,4}
Then the relation R={ (1,1), (2,2), (3,3), (4,4)} is reflexive relation.
6. Equivalence Relation:
A relation R in a set A is said to be Equivalence relation if and only if
the relation is Reflexive, Transitive and Symmetric.
So this was an article on Relation and its Type hope you all like this.
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