Maharashtra State BoardHSC Arts 11th
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Answer the following: Show that the relation R in the set A = {1, 2, 3, 4, 5} Given by R = {(a, b)/|a − b| is even} is an equivalence relation. - Mathematics and Statistics

Sum

Answer the following:

Show that the relation R in the set A = {1, 2, 3, 4, 5} Given by R = {(a, b)/|a − b| is even} is an equivalence relation.

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Solution

R = {(a, b)/|a − b| is even, a, b ∈ A}, where

A = {1, 2, 3, 4, 5}

|a − a| = 0 is even

∴ aRa ∀ a ∈ A

∴ R is reflexive

Let aRb

∴ |a − b| is even

∴ |a − b| = |b − a|

∴ |b − a| is even

∵ bRa

∴ aRb ⇒ bRa ∀a, b ∈ A

∴ R is symmetric

Let aRb and bRc

∴ |a − b| and |b − c| are even

If b is even, then a and c both are even

∴ |a − c| is even

If b is odd, then a and c both are odd

∴ |a − c| is even

∴ aRb, bRc ⇒ aRc ∀a, b, c ∈ A

∴ R is transitive

∵ R is reflexive, symmetric, and transitive

∴ R is an equivalence relation

  Is there an error in this question or solution?
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 5 Sets and Relations
Miscellaneous Exercise 5 | Q II. (11) | Page 105
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