Answer:
A =
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
(i)
R = {(a, b) : |a ? b| is a multiple of 4}
R = {(0, 0),
(0, 4) (4, 0), (0, 8), (8, 0), (0, 12), (12, 0), (1, 1), (1, 5), (5, 1) (1, 9),
(9, 1), (2, 2), (2, 6), (6, 2), (2, 10), (10, 2), (3, 3), (3, 7), (7, 3), (3,
11), (11, 3), (11, 3), (4, 4), (4, 8), (8, 4), (4, 12), (12, 4), (5, 5), (5,
9), (9, 5), (6, 6), (6, 10), (10, 6), *7, 7), (7, 11), (1, 7), (8, 8), (8, 12),
(9, 9), (10, 10), (11, 11), (12, 12)}.
Reflexive
: As |a ? a| = 0 is divisible by 4
Therefore
(a, a)
is reflexive.
Symmetric
: Let (a, b)
is multiple
of 4
|? (b ? a)|is
multiple of 4
is multiple of
4
is symmetric.
Transitive
: (a, b)
and (b, c)
| a ? b| and
|b ? c| are multiple of 4
a ? b and b ?
c are multiple of 4
a ? a + b ? c
is multiple of 4
a ? c is
multiple of 4
|a ? c | is
multiple of 4
(a,
c)
R is
transitive.
Hence
R is an equivalence relation.
Let
B be the set of elements related to 1.
is multiple of
4}
(ii)
{(0, 0), (1,
1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6),
(7,
7), (8 8), (9, 9), (10, 10), (11, 11), (12, 12)}.
Reflexive
: As a =
Symmetric
: Let (a, b)
is symmetric.
Transitive
: (a, b)
and (b, c)
a = b and b
=c
a = c
(a, c)
R is
transitive.
Hence
R is an equivalence relation.
Let
C be the set of elements related to 1
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