question_answer 8)

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 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:

A = {1, 2, 3, 4, 5}       R = {(a, b) : |a ? b| is even}       ,       (1, 3), (3, 1), (1, 5), (5, 1),(2, 4), (4, 2), (3, 5), (5, 3)}       Reflexive : As |a ? a| = 0 (an even number)              R is reflexive.       Symmetric : (a, b) is even                          is symmetric.                     Transitive : (a, b) and (b, c)        is even and |b ? c| is even        a ? b and b ? c are even        (a ? b) + (b ? c) is even        a ? c is even        |a ? c| is also even  (a, c)       is transitive.       Hence R is an equivalence relation.       As |1 ? 3|, |3 ? 1|; |3 ? 5|, |5 ? 3| and |1 ? 5|, |5 ? 1|, are even numbers, therefore all elements of {1, 3, 5} are related to each other.       Also, |2 ? 4| and |4 ? 2| are even number, therefore all elements of {2, 4} are related to each other.       Clearly, |1 ? 2|, |2 ? 1|; |1 ? 4|, |4 ? 1|; |3 ? 2|, |2 ? 3|;       |5 ? 2|; |5 ? 2|, |2 ? 5|; |5 ? 4|, |4 ? 5| are not even        no element of the {1, 3, 5} is relatd to the any element of {2, 4}.