The document defines a binary operation on the set X = {a,b,c,d} according to a multiplication table. It then computes (ca)b and c(ab) according to the table, finding that they are not equal. From this, it concludes that the binary operation is not associative.

1. Let X = {a,b,c,d} and let be the binary operation on X given by the following table:
* a b c d
a d c a b
b c b a c
c a a d c
d b d c a
(d) Compute (ca)b and c(ab). Can you tell, based on this computation, whether is associative?
Explain.
Solution
Let X = {a,b,c,d} and let be the binary operation on X given by the following table:
a b c d
a d c a b
b c b a c
c a a d c
d b d c a
(c*a)*b
c*a = a
a*b = c
(c*a)*b = a*b = c
and c(ab)
a*b = c
c*c = d
c*(a*b) = c*c = d
(ca)b not equal c(ab)

2. it is not asosiative