1.

Consider a binary operation on Q – {1}, defined by a * b = a + b – ab.(i) Find the identity element in Q – {1}.(ii) Show that each a ∈ Q – {1} has its inverse.

Answer»

(i) Consider e as an identity element.

We know that a * e = a Ɐ a ∈ Q {1}

So a + e – ae = a

It can be written as

e (1 – a) = 0

So e = 0 ∈ Q {1}

a * 0 = a + 0 = a

So 0 * a = 0 + a = a

Hence, 0 is the identity element in Q – {1}

(ii) Consider a ∈ Q – {1} where a -1 = b

We know that a * b = 0

It can be written as

a + b – ab = 0

So a = ab – b

a = (a – 1) b

We get b = a/ a – 1 ∈ Q – {1}

So a -1 = a/ a – 1 ∈ Q – {1}

Therefore, each a ∈ Q – {1} has its inverse.


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