April 2026

Let A = R0 x 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 x R. i. Show that ‘O’ is commutative and associative on A ii. Find the identity element in A iii. Find the invertible elements in A

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Binary Operation on Ordered Pairs 📺 Watch Video Explanation: Given: \( (a,b) \circ (c,d) = (ac, bc + d) \) i. Commutativity: \( (a,b)\circ(c,d) = (ac, bc + d) \) \( (c,d)\circ(a,b) = (ca, da + b) \) Since: \( bc + d \neq da + b \ \text{(in general)} \) ❌ NOT commutative Associativity: […]

Let A = R0 x 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 x R. i. Show that ‘O’ is commutative and associative on A ii. Find the identity element in A iii. Find the invertible elements in A Read More »

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}. iii. Show that every element of Q – {-1}. Is invertible. Also, find the inverse of an arbitrary element.

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Binary Operation Full Solution 📺 Watch Video Explanation: Given: \( a*b = a + b + ab, \quad a,b \in \mathbb{Q} \setminus \{-1\} \) i. Commutativity: \( a*b = a + b + ab = b + a + ba = b*a \) ✔ Commutative Associativity: \( (a*b)*c = a + b + c +

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}. iii. Show that every element of Q – {-1}. Is invertible. Also, find the inverse of an arbitrary element. Read More »

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. iii. Find the invertible elements in Z

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Binary Operation Properties 📺 Watch Video Explanation: Given: \( a * b = a + b – 4, \quad a,b \in \mathbb{Z} \) i. Commutativity: \( a*b = a + b – 4 = b + a – 4 = b*a \) ✔ Commutative Associativity: LHS: \( (a*b)*c = (a + b – 4)*c =

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. iii. Find the invertible elements in Z Read More »

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.

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Binary Operation Properties 📺 Watch Video Explanation: Given: \( a * b = \frac{3ab}{5}, \quad a,b \in \mathbb{Q}_0 \) Commutativity: \( a*b = \frac{3ab}{5} = \frac{3ba}{5} = b*a \) ✔ Commutative Associativity: LHS: \( (a*b)*c = \left(\frac{3ab}{5}\right)*c = \frac{3\cdot \frac{3ab}{5} \cdot c}{5} = \frac{9abc}{25} \) RHS: \( a*(b*c) = a*\left(\frac{3bc}{5}\right) = \frac{3a \cdot \frac{3bc}{5}}{5} =

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. Read More »

On the set Z of integers, if the binary operation * is defined by a*b = a + b + 2, then find the identity element.

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Identity Element 📺 Watch Video Explanation: Find identity element Given: \( a * b = a + b + 2, \quad a,b \in \mathbb{Z} \) Identity Definition: \( a * e = a = e * a \) Find \( e \): \( a * e = a + e + 2 = a \)

On the set Z of integers, if the binary operation * is defined by a*b = a + b + 2, then find the identity element. Read More »

If the binary operation * on the set Z is defined by a*b = a + b – 5, then find the identity element with respect to *.

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Identity Element 📺 Watch Video Explanation: Find identity element Given: \( a * b = a + b – 5, \quad a,b \in \mathbb{Z} \) Identity Definition: \( a * e = a = e * a \) Find \( e \): \( a * e = a + e – 5 = a \)

If the binary operation * on the set Z is defined by a*b = a + b – 5, then find the identity element with respect to *. Read More »

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

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Identity Element 📺 Watch Video Explanation: Find identity element Given: \( a * b = a + b + ab, \quad a,b \in \mathbb{Q} \setminus \{-1\} \) Identity Definition: \( a * e = a = e * a \) Find \( e \): \( a * e = a + e + ae =

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

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

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Identity Element 📺 Watch Video Explanation: Find identity element Given: \( a * b = a + b, \quad a,b \in I^+ \) Identity Definition: An element \( e \) is identity if: \( a * e = a = e * a \) Check: \( a * e = a + e = a

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

Let S be the set of all rational numbers except 1 and * be defined on S by a*b = a + b – ab, for all a, b ∈S. Prove that: i. * is a binary operation on S ii. * is commutative as well as associative.

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Binary Operation Full Proof 📺 Watch Video Explanation: Given: \( S = \mathbb{Q} \setminus \{1\}, \quad a*b = a + b – ab \) i. Closure (Binary Operation): \( a*b = a + b – ab = 1 – (1-a)(1-b) \) If \( a, b \neq 1 \), then \( (1-a) \neq 0 \), \(

Let S be the set of all rational numbers except 1 and * be defined on S by a*b = a + b – ab, for all a, b ∈S. Prove that: i. * is a binary operation on S ii. * is commutative as well as associative. Read More »

On Q, the set of all rational numbers a binary operation * is defined by a∗b = (a+b)/2. Show that * is not associative on Q.

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Not Associative Proof 📺 Watch Video Explanation: Prove that the operation is not associative Given: \( a * b = \frac{a+b}{2}, \quad a,b \in \mathbb{Q} \) Proof (Counterexample Method): Take \( a = 1 \), \( b = 3 \), \( c = 5 \) LHS: \( (a*b)*c = \left(\frac{1+3}{2}\right)*5 = 2*5 \) \( =

On Q, the set of all rational numbers a binary operation * is defined by a∗b = (a+b)/2. Show that * is not associative on Q. Read More »