# Test whether the following relation is (1) reflexive (2) symmetric and (3) transitive R on Z defined by (a,b) \[\in R\Leftrightarrow \left| a-b \right|\le 5.\]

Last updated date: 18th Mar 2023

•

Total views: 303.9k

•

Views today: 3.82k

Answer

Verified

303.9k+ views

Hint: we will have to know about the term reflexive, symmetric and transitive so that we can understand the question. For a relation R in set A. The relation said to be reflexive if (a,a) \[\in \]R for every a \[\in \]A, for symmetric relation if (a,b) \[\in \]R then (b,a) \[\in \]R and for transitive relation if (a,b) \[\in \]R, (b,c) \[\in \]R then (a,c) \[\in \]R.

Complete step-by-step answer:

Given the relation is R on Z defined by (a,b) \[\in R\Leftrightarrow \left| a-b \right|\le 5.\]

Now, we will check the condition of reflexive, symmetric and transitive for the above relation.

Clearly, we can say that the above relation is reflexive as \[\forall a\in Z,(a,a)\in R\text{ since }\left| a-a \right|=0\le 5.\]

\[\begin{align}

& \text{Also the relation is symmetric as }\left| b-a \right|=\left| a-b \right|\le 5\text{ so (a,b)}\in R,\forall a,b\in Z. \\

& \text{But the relation is not transitive as (1,2) }\in \text{R,(2,7)}\in \text{R but (1,7)}\notin \text{R}\text{.} \\

\end{align}\]

Therefore, the above given relation is reflexive, symmetric but not transitive.

NOTE: Just remember the term reflexive, symmetric and transitive so you can easily understand the given question and solve it easily. The condition for the relation to be reflexive, symmetric and transitive are mentioned in the hint.

Also, remember the point that if any relation is symmetric,reflexive as well as transitive then the relation is known as equivalence relation.In many questions we have to find the equivalence relation also so it is very important to remember this point.

Complete step-by-step answer:

Given the relation is R on Z defined by (a,b) \[\in R\Leftrightarrow \left| a-b \right|\le 5.\]

Now, we will check the condition of reflexive, symmetric and transitive for the above relation.

Clearly, we can say that the above relation is reflexive as \[\forall a\in Z,(a,a)\in R\text{ since }\left| a-a \right|=0\le 5.\]

\[\begin{align}

& \text{Also the relation is symmetric as }\left| b-a \right|=\left| a-b \right|\le 5\text{ so (a,b)}\in R,\forall a,b\in Z. \\

& \text{But the relation is not transitive as (1,2) }\in \text{R,(2,7)}\in \text{R but (1,7)}\notin \text{R}\text{.} \\

\end{align}\]

Therefore, the above given relation is reflexive, symmetric but not transitive.

NOTE: Just remember the term reflexive, symmetric and transitive so you can easily understand the given question and solve it easily. The condition for the relation to be reflexive, symmetric and transitive are mentioned in the hint.

Also, remember the point that if any relation is symmetric,reflexive as well as transitive then the relation is known as equivalence relation.In many questions we have to find the equivalence relation also so it is very important to remember this point.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE