Question

Let A be Q\{1}. Define * on A by x * y = x + y – xy. Is * binary on A? If so, examine the existence of an identity, the existence of inverse properties for the operation * on A

Answer 1

To verify the identity property:

Let a ∈ A (a ≠ 1)

If possible let e ∈ A such that

a * e = e * a = a

To find e:

a * e = a

i.e. a + e – ae = a

⇒ `"e"(1 - "a")` = 0

⇒ e = `0/(1 - "a")` = 0  ......(∵ a ≠ 1)

So, e = (≠ 1) ∈ A

i.e. Identity property is verified.

To verify the inverse property:

Let a ∈ A  .....(i.e. a ≠ 1)

If possible let a’ ∈ A such that

To find a’:

a * a’ = e

i.e. a + a’ – aa’ = 0

⇒ a'(1 – a) = – a

⇒ a' = `(- "a")/(1 - "a")`

= ``"a"/("a" - 1)` ∈ A  ......(∵ a ≠ 1)

So, a' ∈ A

⇒ For every ∈ A there is an inverse a’ ∈ A such that

a* a’ = a’ * a = e

⇒ Inverse property is verified.