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प्रश्न
Let A be Q\{1} Define * on A by x * y = x + y – xy. Is * binary on A? If so, examine the commutative and associative properties satisfied by * on A
उत्तर
Let a, b ∈ A
i.e. a ≠ ±1 , b ≠ 1
Now a * b = a + b – ab
If a + b – ab = 1
⇒ a + b – ab – 1 = 0
i.e. a(1 – b) – 1(1 – b) = 0
(a – 1)(1 – b) = 0 ⇒ a = 1, b = 1
But a ≠ 1 , b ≠ 1
So (a – 1)(1 – 6) ≠ 1
i.e. a * b ∈ A.
So * is a binary on A.
To verify the commutative property:
Let a, b ∈ A
i.e. a ≠ 1, b ≠ 1
Now a * b = a + b – ab
And b * a = b + a – ba
So a * b = b * a
⇒ * is commutative on A.
To verify the associative property:
Let a, b, c ∈ A
i.e. a, b, c ≠ 1
To prove the associative property we have to prove that
a * (b * c) = (a * b) * c
L.H.S: b * c = b + c – bc = D .......(say)
So a * (b * c) = a * D = a + D – aD
= a + (b + c – bc) – a(b + c – bc)
= a + b + c – bc – ab – ac + abc
= a + b + c – ab – bc – ac + abc .......(1)
R.H.S: (a * b) = a + b – ab = K ......(say)
So (a * b) * c = K * c = K + c – Kc
= (a + b – ab) + c – (a + b – ab)c
= a + b – ab + c – ac – bc + abc
= a + b + c – ab – bc – ac + abc ......(2)
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