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}.
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