question_answer 9)

Show that each of the relation R in the set  given by        R = {(a, b) : |a ? b| is a multiple of 4} (ii)    R = {(a, b) : a = b}       is an equivalence relation. Find the set of all elements related to 1 in each case.  

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