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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
Solution
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.
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