Question

Consider the binary operations*: R ×R → and o: R × R → R defined as and a o b = a, &mnForE;a, b ∈ R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that &mnForE;a, b, c ∈ R, a*(b o c) = (a * b) o (a * c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.

Answer 1

It is given that *: R ×R → and o: R × R → R isdefined as

and a o b = a, &mnForE;a, b ∈ R.

For a, b ∈ R, we have:

∴a * b = b * a

∴ The operation * is commutative.

It can be observed that,

∴The operation * is not associative.

Now, consider the operation o:

It can be observed that 1 o 2 = 1 and 2 o 1 = 2.

∴1 o 2 ≠ 2 o 1 (where 1, 2 ∈ R)

∴The operation o is not commutative.

Let a, b, c ∈ R. Then, we have:

(a o b) o c = a o c = a

a o (b o c) = a o b = a

⇒ a o b) o c = a o (b o c)

∴ The operation o is associative.

Now, let a, b, c ∈ R, then we have:

a * (b o c) = a * b =

(a * b) o (a * c) =

Hence, a * (b o c) = (a * b) o (a * c).

Now,

1 o (2 * 3) =

(1 o 2) * (1 o 3) = 1 * 1 =

∴1 o (2 * 3) ≠ (1 o 2) * (1 o 3) (where 1, 2, 3 ∈ R)

The operation o does not distribute over *.