Home
Class 12
MATHS
Prove that the relation R in set A = {1,...

Prove that the relation R in set A = {1, 2, 3, 4, 5} given by R = {(a,b): |a-b| is even} is an equivalence relation .

Text Solution

Verified by Experts

`A= {1,2,3,4,5}`
`R={(a,b):|a-b|` is even }
It is clear that for any clement `a in A,` we have `|a - a| = 0`
(which is even).
Therefore, R is reflexive.
Let `(a,b) in R`.
`implies |a-b|` is even,
` implies |-(a-b)|=|b-a|` is also even
`implies (b,a) in R`
Therefore, R is symmetric.
Now, let `(a,b) in R ` and `(b,c) in R.`
` implies |a-b|` is even and `|b-c|` is even
`implies (a-b)` is even and `(b-c)` is even ` " " ` (assuming that `a gt b gt c`)
`implies (a-c)=(a-b)+(b-c)` is even ` " " ` [Sum of two even integers is even]
` implies |a-c|` is even
`implies (a,c) in R`
Therefore, R is transitive.
Hence, R is an equivalence relation.
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise Solved Examples|15 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 1.1|15 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE ENGLISH|Exercise Archives (Numerical Value Type)|3 Videos

  • SCALER TRIPLE PRODUCTS

    CENGAGE ENGLISH|Exercise DPP 2.3|11 Videos

Similar Questions

Explore conceptually related problems

Show that the relation R on the set A={1,\ 2,\ 3,\ 4,\ 5} , given by R={(a ,\ b):|a-b| is even }, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But, no element of {1, 3, 5} is related to any element of {2, 4}.

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a , b) : |a - b| is e v e n} , is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4 } are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4} .

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is divisible by 2} is an equivalence relation. Write all the equivalence classes of R .

Show that the relation R on the set A={x in Z ;0lt=xlt=12} , given by R={(a ,\ b): a=b} , is an equivalence relation. Find the set of all elements related to 1.

Show that the relation R on the st A={ xin Z:0lexle12} , given by R={a,b),|a-b| is multiple of 4} is an equivalence relation.

Show that the relation R on the set A{xZ ;0lt=12}, given by R={(a , b): a=b}, is an equivalence relation. Find the set of all elements related to 1.

Show that the relation R on the set Z of integers, given by R={(a ,\ b):2 divides a-b} , is an equivalence relation.

Show that the relation R on the set Z of integers, given by R={(a ,\ b):2 divides a-b} , is an equivalence relation.

The relation R defined on the set A = {1, 2, 3, 4, 5} by R = {(a, b): |a^(2)-b^(2)|lt16} is given by

Show that the relation R on the set A={x in Z :0lt=xlt=12} , given by R={(a ,\ b):|a-b| is a multiple of 4} is an equivalence relation. Find the set of all elements related to 1 i.e. equivalence class [1].