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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. 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 \(\{2, 4\}\).

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  • A relation R in a set A is an equivalence relation if R is reflexive, symmetric and transitive.
  • A relation R in a set A is called reflexive. if $(a,a) \in R\;for\; all\; a\in A$
  • A relation R in a set A is called symmetric. if $(a_1,a_2) \in R\;\Rightarrow \; (a_2,a_1)\in R \;$ for $\;a_1,a_2 \in A$
  • A relation R in a set A is called transitive. if $(a_1,a_2) \in\; R$ and $(a_2,a_3)\in R \Rightarrow \;(a_1,a_3)\in R\; $for all $\; a_1,a_2,a_3 \in A$
Given the set $A=\{1,2,,3,4,5\}$ and the relation $R=\{(a,b):|a-b| \;is\; even\}$:
Let $a=b$, $(a,a) \in R \rightarrow |a-a|=0$ which is even. Therefore $R$ is reflexive.
For $R$ to be symmetric, if $(a,b) \in R \rightarrow (b,a) \in R$.
$(a,b) \in R \rightarrow |a-b|=even$
$(b,a) \in R \rightarrow |b-a|=even$
$(a-b)=-(b-a); $ therefore $|(a-b)|=|(b-a)|$
Therefore $(b,a)\in R$. Hence, $R$ is symmetric
Let $(a,b) \in R \; and \;(b,c)\in R$
$\Rightarrow |a-b|\; is\; even$ and $|b-c|\;is \;even$
If $(a,c) \in R \rightarrow |a-c| = even$
Now, $a-c=a-b+b-c$, which is an even number, as the sum of even numbers is even.
Hence $R$ is transitive.
Since $R$ is reflexive, symmetric and transitive. $R$ is an eqivalence relation.
To show all elements of subset are related to each other, we need to show that the difference of any two elements in the set is even.
As all elements of the subset {1,3,5} are odd numbers, the difference between any two odd numbers results in even number.
Therefore given a subset {1,3,5}, all elements in this set are related to each other.
Similarly, since the difference of two even numbers is even, all elements in the subset {2,4} are also related to each other
Since the difference of an odd number and even number is not even number, No elements of {1,3,5} can be related to any elements of {2,4}
answered Feb 21, 2013 by meena.p
edited Mar 8, 2013 by balaji.thirumalai

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