A relation \(R\) from a non-empty set \(A\) to a non-empty set \(B\) is a subset of the cartesian product \(A \times B\).
It "maps" elements of one set to another set. The subset is derived by describing a relationship
between the first element and the second element of the ordered pair \(\left( {A \times B} \right)\). Domain: The set of all first elements of the ordered pairs in a relation \(R\) from a set
\(A\) to a set \(B\) is called the domain of the relation \(R\). Range: the set of all the ending points is called the range
Types Of Relations
Important Note: Let A be a set. Then we can form a relation from A to A. Instead of calling such a relation "a relation from A to A" we instead say that R is a "relation on A".
Empty Relation
A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., $R = \phi \subset A \times A$.
Universal Relation
A relation R in a set A is called universal relation if all elements of A is related to every element of A i.e R= A X A.
Reflexive Relation
A relation in a set A is called reflexive relation if (a,a) ∈ R for every element a ∈ A. Example:.
Let A = {1, 2, 3, 4,5,6,7,8,9,10} and define R = {(a, b) | a divides b}
We saw that R was reflexive since every number divides itself
Let A = {1, 2, 3, 4,5,6,7,8,9,10} and define R ={(1,1),(2,2),(2,3),(3,2),(4,4)}
We saw that R is not reflexive since every number is not present in R
Symmetric Relation
A relation in a set A is called if (a,b) ∈ R the (b,a) ∈ R for all a,b ∈ A Example
Let A = {1, 2, 3, 4} and R = {(1, 2), (2,1),(3,3),(4,4)}
This relation is symmetric. It satisfies the above criterion. Important Note:
symmetry is a different kind of requirement than reflexivity. Reflexivity requires that certain pairs must be in R, namely all pairs of the form (a, a) for every element in A.
However symmetry only requires that if a pair (a, b) is in R, then the pair (b, a) must also be in R. But it is not required that pairs of the form (a, b) are in R unless the pair (b, a) is in R. Simply stated, you must have both pairs or neither
Transitive Relation
A relation R on a set A is called transitive if whenever (a, b) is in R and (b, c) is in R, then (a, c) is in R. Example :
Let A = Z and define R = {(a, b) | a > b}.
R is transitive because if a > b and b > c then a > c. Important Note:
Note that transitivity, like symmetry, is possessed by a relation unless the stated condition is violated. So unless you can find pairs (a, b) and (b, c) which are in R while (a, c) is not, then the relation is transitive. In particular, the empty relation is always transitive because it has no pairs to violate the condition
Equivalence Relation
A relation R on a Set A is called equivalence relation if R is reflexive,Symmetric and transistive Example :
Let A = Z and define R = {(a, b) | 3 divides a - b}.
It is reflexive as (a-a) will always be divided by 3
It is symmetric as if (a-b) will be divided by 3 ,the (b-a) will also divided by 3
it is transitive as if (a,b) ∈ R and (b,) ∈ R which means a-b and b-c are divided by 3, now a-c=(a-b)+(b-c) ,so (a,c) ∈ R
Solved Examples
Question 1
Determine if the below relation is an equivalence Relation
A={1,2,3,4,5,.....,14)
R is the relation on set A defined as
R={(a,b)| (3a-b)=0}
Solution :
From the defination , R would contain following elements
R={(1,3),(2,6),(3,9),(4,12)}
R is not reflexive as we dont have (1,1),(2,2) like that in the Relation
R is not symmetric also as we dont (3,1) for (1,3) in the relation
R is not transitive also as dont have (1,9) for (1,3), (3,9) in R
Question 2
Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),(3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive. Solution :
R is reflexive, since (1, 1), (2, 2) and (3, 3) lie in R.
Also, R is not symmetric, as (1, 2) ∈ R but (2, 1) ∉ R.
Similarly, R is not transitive, as (1, 2) ∈ R and (2, 3) ∈ R but (1, 3) ∉ R
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