Identity relation is the one in which every elements maps to itself only.
The rule for identity relation is given below.
"Every element is related to itself only"
Let R be a relation defined on the set A
If R is identity relation, then
R = {(a, a) / for all a ∈ A}
That is, every element of A has to be related to itself only.
In case, there is an ordered pair (a, b) in R, then R is not identity. Because the ordered pair (a, b) does not satisfy the above rule of identity relation.
Example 1 :
Let A = {1, 2, 3} and R be a relation defined on set A as
R = {(1, 1), (2, 2), (3, 3)}
Verify R is identity.
Solution :
In the set A, we find three elements. They are 1, 2 and 3.
When we look at the ordered pairs of R, we find the following associations.
(1, 1) -----> 1 is related to 1
(2, 2) -----> 2 is related to 2
(3, 3) -----> 3 is related to 3
In R, every element of A is related to itself and not to any other different element.
That is, every element of A is related to itself only.
So, R is identity.
Example 2 :
Let A = {1, 2, 3} and R be a relation defined on set A as
R = {(1, 1), (2, 2), (3, 3), (2, 3)}
Verify R is identity.
Solution :
In the set A, we find three elements. They are 1, 2 and 3.
When we look at the ordered pairs of R, we find the following associations.
(1, 1) -----> 1 is related to 1
(2, 2) -----> 2 is related to 2
(3, 3) -----> 3 is related to 3
(2, 3) -----> 2 is related to 3
In R, every element of A is related to itself and also the element "2" is related to a different element "3".
Here, we can not say that every element of A is related to itself only.
So, R is not identity.
The two relations reflexive and identity appear, as if they were same.
But, there is a huge difference between them.
The difference between reflexive and identity relation can be described in simple words as given below.
Reflexive : Every element is related to itself
Identity : Every element is related to itself only
Let us consider an example to understand the difference between the two relations reflexive and identity.
Let A = {1, 2, 3}.
Let R1 and R2 be two relations defined on set A such that
R1 = {(1,1), (2,2), (3,3), (1,2)}
R2 = {(1,1), (2,2), (3,3)}
When we look at R1, every element of A is related to itself and also, the element "1" is related to a different element "2".
More details about R1 :
(i) "1" is related to "1", "2" is related to "2" and "3" is related to "3"
(ii) Apart from "1" is related to "1", "1" is also related to "2"
Here we can not say that "1" is related to "1" only.
Because "1" is related to "2" also.
This is the point which makes the reflexive relation to be different from identity relation.
Hence R1 is reflexive relation.
When we look at R2, every element of A is related to it self and no element of A is related to any different element other than the same element.
More details about R2 :
(i) "1" is related to "1", "2" is related to "2" and "3" is related to "3"
(ii) "1" is related to "1" and it is not related to any different element.
The same thing happened to "2" and "3".
(iii) From the second point, it is very clear that every element of R is related to itself only. No element is related to any different element
This is the point which makes identity relation to be different from reflexive relation.
Hence R2 is identity relation.
That is,
Reflexive : Every element is related to itself
Identity : Every element is related to itself only
Difference between reflexive and identity relation
Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.
If you have any feedback about our math content, please mail us :
v4formath@gmail.com
We always appreciate your feedback.
You can also visit the following web pages on different stuff in math.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Trigonometry word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits