Question

Problem 1.2 Let A = {a,b,c). An operation on A is defined by the following table: a b c db blb c Q c'c a b It means that aa=a, a ob=d, a oc=b, boa=b etc Show that the table does not define a binary operation on A.​

Answer 1

Answer:

Problem 1.2 Let A = {a,b,c). An operation on A is defined by the following table: a b c db blb c Q c'c a b It means that aa=a, a ob=d, a oc=b, boa=b etc Show that the table does not define a binary operation on A.

Answer 2

Answer:

A=Q×Q . [Given]

For any (a,b),(c,d)∈A, ∗ is defined by

(a,b)∗(c,d)=(ac,b+ad) ... [Given]

To check ∗ is commutative i.e. to check (a,b)∗(c,d)=(c,d)∗(a,b) for any (a,b),(c,d)∈A

Now, (a,b)∗(c,d)=(ac,b+ad)

(c,d)∗(a,b)=(ca,d+cb)=(ac,d+bc)



=(ac,b+ad)

∴(a,b)∗(c,d)



=(c,d)∗(a,b)

Thus, ∗ is not commutative ... (1)