RD Sharma

For the binary operation ×10 set S = {1, 3, 7, 9}, find the inverse of 3.

Answer: 1 ×10 1 = remainder obtained by dividing 1 × 1 by 10 = 1 3 ×10 7 = remainder obtained by dividing 3 × 7 by 10 = 1 7 ×10 9 = remainder obtained by dividing 7 × 9 by 10 = 3 Composition table:...

Construct the composition table for ×5 on set Z5 = {0, 1, 2, 3, 4}

Answer: 1 ×5 1 = remainder obtained by dividing 1 × 1 by 5 = 1 3 ×5 4 = remainder obtained by dividing 3 × 4 by 5 = 2 4 ×5 4 = remainder obtained by dividing 4 × 4 by 5 = 1 Composition table: ×5 0 1...

Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.

Answer: 1 ×6 1 = remainder obtained by dividing 1 × 1 by 6 = 1 3 ×6 4 = remainder obtained by dividing 3 × 4 by 6 = 0 4 ×6 5 = remainder obtained by dividing 4 × 5 by 6 = 2 Composition table: ×6 0 1...

Construct the composition table for +5 on set S = {0, 1, 2, 3, 4}

Answer: 1 +5 1 = remainder obtained by dividing 1 + 1 by 5 = 2 3 +5 1 = remainder obtained by dividing 3 + 1 by 5 = 2 4 +5 1 = remainder obtained by dividing 4 + 1 by 5 = 3 Composition Table: +5 0 1...

Construct the composition table for ×4 on set S = {0, 1, 2, 3}.

Answer: Given, ×4 on set S = {0, 1, 2, 3} 1 ×4 1 = remainder obtained by dividing 1 × 1 by 4 = 1 0 ×4 1 = remainder obtained by dividing 0 × 1 by 4 = 0 2 ×4 3 = remainder obtained by dividing 2 × 3...

Let * be a binary operation on Z defined by a * b = a + b – 4 for all a, b ∈ Z. (i) Show that * is both commutative and associative. (ii) Find the identity element in Z

Answers: (i) Consider, a, b ∈ Z a * b = a + b – 4 = b + a – 4 = b * a a * b = b * a, ∀ a, b ∈ Z Then, * is commutative on Z. a * (b * c) = a * (b + c – 4) = a + b + c -4 – 4 = a + b + c – 8 (a * b)...

Let * be a binary operation on Q0 (set of non-zero rational numbers) defined by a * b = (3ab/5) for all a, b ∈ Q0. Show that * is commutative as well as associative. Also, find its identity element, if it exists.

Answer: Consider, a, b ∈ Q0 a * b = (3ab/5) = (3ba/5) = b * a a * b = b * a, for all a, b ∈ Q0   a * (b * c) = a * (3bc/5) = [a (3 bc/5)] /5 = 3 abc/25 (a * b) * c = (3 ab/5) * c = [(3 ab/5)...

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, ∀ a, b ∈ Q – {-1}   a * (b * c) = a * (b + c + b c) = a + (b + c + b c) + a (b + c + b c) = a + b +...

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}...

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

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

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

Answer: Consider, F = (m, n) be the inverse in A ∀ m ∈ R0 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 =...

Let * be a binary operation on Z defined by a * b = a + b – 4 for all a, b ∈ Z. Find the invertible element in Z.

Answer: Consider, a ∈ Z and b ∈ Z be the inverse of a. a * b = e = b * a a * b = e and b * a = e a + b – 4 = 4 and b + a – 4 = 4 b = 8 – a ∈ Z Hence, 8 – a is the inverse of a ∈...

Find the identity element in the set of all rational numbers except – 1 with respect to * defined by a * b = a + b + ab

Answer: Consider, e be the identity element in I+ with respect to * such that a * e = a = e * a, ∀ a ∈ Q – {-1} a * e = a and e * a = a, ∀ a ∈ Q – {-1} a + e + ae = a and e + a + ea = a, ∀ a ∈ Q –...

Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.

Answer: Consider, e be the identity element in I+ with respect to * a * e = a = e * a, ∀ a ∈ I+ a * e = a and e * a = a, ∀ a ∈ I+ a + e = a and e + a = a, ∀ a ∈ I+ e = 0, ∀ a ∈ I+ Hence, 0 is the...

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