EQUIVALENCE RELATION
About "Equivalence relation"
Equivalence relation :
As we have rules for reflexive, symmetric and transitive relations, we don't have any specific rule for equivalence relation.
Let R be a relation defined on a set A.
If the three relations reflexive, symmetric and transitive hold in R, then it is an equivalence-relation.
To verify equivalence, we have to check whether the three relations reflexive, symmetric and transitive hold.
Apart from the equivalence relation, we have some other relations on sets.
Let us discuss the other different types of relations here:
The other different types of relations on sets are
(i) Reflexive relation
(ii) Symmetric relation
(iii) Transitive relation
(iv) Identity relation
(v) Inverse relation
Let us discuss the above different types of relations in detail.
Reflexive relation
The rule for reflexive relation is given below.
" Every element is related to itself "
Let R be a reflexive relation defined on the set A and
a ∈ A, then R = { (a, a) / for all a ∈ A }
That is, every element of A has to be related to itself.
Example :
Let A = {1, 2, 3 } and R be a reflexive relation defined on set A.
Then, R = { (1, 1), (2, 2), (3, 3) }
Symmetric relation
Let R be a symmetric relation defined on the set A and
a, b ∈ A, then R = { (a, b), (b, a) / for all a, b ∈ A }
That is, if "a" is related to "b", then "b" has to be related to "a" for all "a" and "b" belonging to A.
Example :
Let A be the set of two male children in a family, R be a relation defined on set A and R = "is brother of". Verify whether R is symmetric.
Let a, b ∈ A.
If "a" is brother of "b", then "b" has to be brother of "a".
Clearly, R = { (a, b), (b, a) }
Hence, R is symmetric.
Transitive relation
Let R be a transitive relation defined on the set A and a, b, c ∈ A,
then R = { (a, b), (b, c), (a, c) / for all a,b,c ∈ A }
That is,
If "a" is related to "b" and "b" is related to "c", then "a" has to be related to "c".
Example :
Let A = { 1, 2, 3 }, R be a relation defined on set A as "is less than" and R = { (1, 2), (2, 3), (1,3) } Verify R is transitive.
From the given set A, let a = 1, b = 2 and c = 3.
Then, we have
(a, b ) = (1, 2) --> 1 is less than 2
(b, c ) = (2, 3) --> 2 is less than 3
(a, c ) = (1, 3) --> 1 is less than 3
That is, if 1 is less than 2 and 2 is less than 3, then 1 is less than 3
Clearly, the above points prove that R is transitive.
Important note :
For a particular ordered pair in R, if we have (a, b) and we don't have (b, c), then we don't have to check transitive for that ordered pair.
So, we have to check transitive, only if we find both (a, b) and (b, c) in R.
If the above three relations reflexive, symmetric and transitive hold, then it is equivalence relation.
Identity relation
The rule for identity relation is given below.
" Every element is related to itself only "
Let R be an identity relation defined on the set A and
a ∈ A, then R = { (a, a) / for all a ∈ A }
That is, every element of A has to be related to itself only.
Example :
Let A = {1, 2, 3 } and R be an identity relation defined on set A.
Then, R = { (1, 1), (2, 2), (3, 3) }
Here, every element of A is related to it self and not to any other different element.
Inverse relation
Let R be a relation defined on the set A, a, b ∈ A and R = { (a, b) }, then the inverse relation of R is
R⁻¹ = { (b, a) }
In the given relation, if "a" is related to "b", then in the inverse relation "b" will be related to "a".
Example :
Let R = { (1, 2), (2, 3), (7, 5) }
Then R⁻¹ = { (2, 1), (3, 2), (5, 7) }
Difference between reflexive and identity relation
Reflexive = " Every element is related to itself "
Identity = " Every element is related to itself only "
Let us consider an example to have better understanding of the difference between the two relations. (Reflexive vs 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
Important terms related to relations
If a relation does mapping from the set A to set B, then we can define the following terms.
Domain : Set A
Co domain : Set B
Range : Elements of B involved in mapping.
Let R and R⁻¹ be the two relations which are inverse to each other.
Then we have,
Domain (R) = Range (R⁻¹)
Range (R) = Domain (R⁻¹)
After having gone through the stuff given above, we hope that the students would have understood "Equivalence relation".
Apart from the stuff given above, if you want to know more about "Equivalence relation", please click here
Apart from "Equivalence relation", if you need any other stuff in math, please use our google custom search here.
WORD PROBLEMS
HCF and LCM 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
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
