# REFLEXIVE RELATION

Reflexive relation is the one in which every element maps to itself.

The rule for reflexive relation is given below.

"Every element is related to itself"

Let R be a relation defined on the set A

If R is reflexive relation, 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 relation defined on  set A as

R  = {(1, 1), (2, 2), (3, 3)}

Verify R is reflexive.

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, it is clear that every element of A is related to itself.

So, R is reflexive relation.

## Practice Problems

Problem 1 :

Let A  =  {2, 3, 7}, R be a relation defined on set as

R  =  {(2, 2), (3, 3), (2, 3) (3, 7)}

Determine whether R is reflexive relation.

Solution :

If the relation on set A is reflexive, every element of A has to be related to itself.

In the set A, we find three elements. They are 1, 2 and 3.

So, we must have

2 has to be related to itself ------> (2, 2)

3 has to be related to itself ------> (3, 3)

7 has to be related to itself ------> (7, 7)

But, the ordered pair (7,7) is not in R.

So, R is not reflexive.

Problem 2 :

Let A  =  {1, 2, 3} and R be a relation defined on set A as

R  = {(1, 1), (2, 2), (3, 3), (1, 2)}

Determine whether R is reflexive relation.

Solution :

If the relation on set A is reflexive, every element of A has to be related to itself.

In the set A, we find three elements. They are 1, 2 and 3.

So, we must have

1 has to be related to itself ------> (1, 1)

2 has to be related to itself ------> (2, 2)

3 has to be related to itself ------> (3, 3)

We have all the three ordered pairs (1, 1), (2, 2) and (3, 3) in R.

So, R is reflexive.

Problem 3 :

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), (2, 3), (3, 2)}

Determine whether R is reflexive relation.

Solution :

If the relation on set A is reflexive, every element of A has to be related to itself.

In the set A, we find three elements. They are 1, 2 and 3.

So, we must have

1 has to be related to itself ------> (1, 1)

2 has to be related to itself ------> (2, 2)

3 has to be related to itself ------> (3, 3)

We have all the three ordered pairs (1, 1), (2, 2) and (3, 3) in R.

So, R is reflexive.

## Difference between Reflexive and Identity Relation

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

## Related Topics

Transitive relation

Equivalence relation

Identity relation

Inverse relation

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

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

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

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

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

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

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

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

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6

Featured Categories

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test 