Question

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R={(a,b) : Ia-bI 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}.

Answer 1

A = {1, 2, 3, 4, 5}
R={(a,b) : Ia-bI is even}
It is clear that for any element a ∈A, we have Ia-aI = 0 (which is even).
∴R is reflexive. 

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Let (a, b) ∈ R.
\(\Rightarrow\) Ia-bI is even.
\(\Rightarrow\) I-(a-b)I=Ib-aI is also even.
\(\Rightarrow\) (b,a)∈R
∴R is symmetric.

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Now, let (a, b) ∈ R and (b, c) ∈ R. 
\(\Rightarrow\) Ia-bI is even and Ib-cI is even.
(a-b)is even and (b-c) is even.
\(\Rightarrow\) (a-c)=(a-b)+(b-c) is even. (sum of two integers is even)
\(\Rightarrow\) Ia-cI is even.
⇒ (a, c) ∈ R
∴R is transitive.

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Hence, R is an equivalence relation.
Now, all elements of the set {1, 2, 3} are related to each other as all the elements of
this subset are odd. Thus, the modulus of the difference between any two elements will
be even.

Similarly, all elements of the set {2, 4} are related to each other as all the elements of
this subset are even.

Also, no element of the subset {1, 3, 5} can be related to any element of {2, 4} as all
elements of {1, 3, 5} are odd and all elements of {2, 4} are even. Thus, the modulus of
the difference between the two elements (from each of these two subsets) will not be
even.