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 R is equivalence relation.
To verify equivalence, we have to check whether the three relations reflexive, symmetric and transitive hold.
To know the three relations reflexive, symmetric and transitive in detail, please click on the following links.
Equivalence Relation - Example
Problem :
Let A = {1, 2, 3} and R be a relation defined on set A as
R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}
Verify R is equivalence.
Solution :
We have to check whether the three relations reflexive, symmetric and transitive hold in R.
Reflexive :
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, it is clear that every element of A is related to itself.
So, R is reflexive relation.
Symmetric :
To verify whether R is transitive, we have to check the condition given below for each ordered pair in R.
That is,
(a, b) -----> (b, a)
Let's check the above condition for each ordered pair in R.
From the table above, it is clear that R is symmetric.
Transitive :
To verify whether R is transitive, we have to check the condition given below for each ordered pair in R.
That is,
(a, b), (b, c) -----> (a, c)
Let's check the above condition for each ordered pair in R.
From the table above, it is clear that R is transitive.
Note :
For the ordered pair (3, 3), we don't find the ordered pair (b, c). So, we don't have to check the condition of transitive relation for that ordered pair.
Conclusion :
For the given relation R, all the three relations reflexive, symmetric and transitive hold.
Hence, R is equivalence.
Related Topics
Difference between reflexive and identity relation
After having gone through the stuff given above, we hope that the students would have understood equivalence relation.
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
You can also visit our 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
Recent Articles
-
Vertices of a Right Triangle Worksheet
Oct 15, 19 10:27 PM
Vertices of a Right Triangle Worksheet - Concept - Problems with step by step explanation
-
Vertices of Rectangle Worksheet
Oct 15, 19 10:07 PM
Vertices of Rectangle Worksheet - Concept - Problems with step by step explanation
-
Angles and Their Measures Worksheet
Oct 15, 19 09:50 PM
Angles and Their Measures Worksheet - Concept - Problems with step by step solutions
