Class 12

Let * be a binary operation on Q – {-1} defined by a * b = a + b + ab for all a, b ∈ Q – {-1}. Then, (i) Show that * is both commutative and associative on Q – {-1} (ii) Find the identity element in Q – {-1}

Answers: (i) Consider, a, b ∈ Q – {-1} a * b = a + b + ab = b + a + ba = b * a a * b = b * a, βˆ€ a, b ∈ Q – {-1}   a * (b * c) = a * (b + c + b c) = a + (b + c + b c) + a (b + c + b c) = a + b +...

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Let * be a binary operation on Q – {-1} defined by a * b = a + b + ab for all a, b ∈ Q – {-1}. Then, Show that every element of Q – {-1} is invertible. Also, find inverse of an arbitrary element.

Answer: Consider, a ∈ Q – {-1} and b ∈ Q – {-1} be the inverse of a. a * b = e = b * a a * b = e and b * a = e a + b + ab = 0 and b + a + ba = 0 b (1 + a) = – a Q – {-1} b = -a/1 + a Q – {-1}...

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Let A = R0 Γ— R, where R0 denote the set of all non-zero real numbers. A binary operation β€˜O’ is defined on A as follows: (a, b) O (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 Γ— R. (i) Show that β€˜O’ is commutative and associative on A (ii) Find the identity element in A

Answers: (i) Consider, X = (a, b) Y = (c, d) ∈ A, βˆ€ a, c ∈ R0 b, d ∈ R X O Y = (ac, bc + d) Y O X = (ca, da + b) X O Y = Y O X, βˆ€ X, Y ∈ A O is not commutative on A. X = (a, b) Y = (c, d) a Z = (e,...

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Let A = R0 Γ— R, where R0 denote the set of all non-zero real numbers. A binary operation β€˜O’ is defined on A as follows: (a, b) O (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 Γ— R. Find the invertible element in A.

Answer: Consider, F = (m, n) be the inverse in A βˆ€ m ∈ R0Β and n ∈ R X O F = E F O X = E (am, bm + n) = (1, 0) and (ma, na + b) = (1, 0) Considering (am, bm + n) = (1, 0) am = 1 m = 1/a And bm + n =...

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