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Check the commutativity and associativity of the binary operations:‘*’ on N defined by a*b = 2^ab ∀ a, b ∈N

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Commutativity and Associativity Check 📺 Watch Video Explanation: Check commutativity and associativity Given: \( a * b = 2^{ab}, \quad a,b \in \mathbb{N} \) Commutativity: \( a * b = 2^{ab} \) \( b * a = 2^{ba} = 2^{ab} \) ✔ Operation is commutative Associativity: LHS: \( (a*b)*c = 2^{(2^{ab})c} \) RHS: \( a*(b*c) […]

Check the commutativity and associativity of the binary operations:‘*’ on N defined by a*b = 2^ab ∀ a, b ∈N Read More »

Check the commutativity and associativity of the binary operations:‘*’ on Z defined by a*b = a + b + ab ∀ a, b ∈Z

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Commutativity and Associativity Check 📺 Watch Video Explanation: Check commutativity and associativity Given: \( a * b = a + b + ab \quad \forall a,b \in \mathbb{Z} \) Commutativity: \( a * b = a + b + ab \) \( b * a = b + a + ba = a + b

Check the commutativity and associativity of the binary operations:‘*’ on Z defined by a*b = a + b + ab ∀ a, b ∈Z Read More »

Let A be any set containing more than one element. Let ‘*’ be a binary operation on A defined by a*b = b for all a, b ∈A. Is ‘*’ commutative or associative on A ?

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Commutativity and Associativity Check 📺 Watch Video Explanation: Check whether the operation is commutative and associative Given: \( a * b = b \quad \forall a,b \in A \) Commutativity: \( a * b = b \) \( b * a = a \) Clearly: \( a * b \neq b * a \quad (\text{when

Let A be any set containing more than one element. Let ‘*’ be a binary operation on A defined by a*b = b for all a, b ∈A. Is ‘*’ commutative or associative on A ? Read More »

Determine which of the following binary operations are associative and which are commutative:* on Q defined by a∗b = (a + b)/2 for all a, b∈ Q.

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Associativity and Commutativity Check 📺 Watch Video Explanation: Determine associativity and commutativity Given: \( a * b = \frac{a + b}{2}, \quad a,b \in \mathbb{Q} \) Commutativity: \( a * b = \frac{a+b}{2} = \frac{b+a}{2} = b * a \) ✔ Operation is commutative Associativity: Check LHS: \( (a*b)*c = \left(\frac{a+b}{2} * c\right) = \frac{\frac{a+b}{2}

Determine which of the following binary operations are associative and which are commutative:* on Q defined by a∗b = (a + b)/2 for all a, b∈ Q. Read More »

Determine the binary operations are associative and which are commutative:* on N defined by a*b=1 ∀ a,b∈ N

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Associativity and Commutativity Check 📺 Watch Video Explanation: Determine associativity and commutativity Given: \( a * b = 1 \quad \forall \, a,b \in \mathbb{N} \) Commutativity: \( a * b = 1 = b * a \) ✔ Operation is commutative Associativity: Check: \( (a*b)*c = 1*c = 1 \) \( a*(b*c) = a*1

Determine the binary operations are associative and which are commutative:* on N defined by a*b=1 ∀ a,b∈ N Read More »

Let * be a binary operation on N given by a*b = LCM(a, b) for all a, b∈ N. Find 5 * 7.

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Evaluate Binary Operation using LCM 📺 Watch Video Explanation: Find the value of \( 5 * 7 \) Given: Binary operation defined by \( a * b = \mathrm{LCM}(a, b) \) Solution: Since 5 and 7 are prime numbers, their LCM is: \( \mathrm{LCM}(5,7) = 5 \times 7 = 35 \) Final Answer: ✔ \(

Let * be a binary operation on N given by a*b = LCM(a, b) for all a, b∈ N. Find 5 * 7. Read More »

Let ‘*’ be a binary operation on N defined by a*b = L.C.M(a,b) for all a, b ∈N. Find (i) 2*4, 3*5, 1*6 (ii)Check the commutativity and associativity of ‘*’ on N.

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LCM Binary Operation Properties 📺 Watch Video Explanation: Given Binary Operation \( a * b = \mathrm{LCM}(a, b) \) (i) Find Values \( 2 * 4 = \mathrm{LCM}(2,4) = 4 \) \( 3 * 5 = \mathrm{LCM}(3,5) = 15 \) \( 1 * 6 = \mathrm{LCM}(1,6) = 6 \) (ii) Commutativity Since: \( \mathrm{LCM}(a,b) =

Let ‘*’ be a binary operation on N defined by a*b = L.C.M(a,b) for all a, b ∈N. Find (i) 2*4, 3*5, 1*6 (ii)Check the commutativity and associativity of ‘*’ on N. Read More »

Let S be the set of all rational numbers of the form m/n, where m∈Z and n = 1,2,3. Prove that * on S defined by a*b = ab is not a binary operation.

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Binary Operation on Rational Set 📺 Watch Video Explanation: Determine whether the operation is a binary operation or not Given: \( S = \left\{ \frac{m}{n} : m \in \mathbb{Z}, \; n = 1,2,3 \right\} \) Operation: \( a * b = ab \) Concept: A binary operation must satisfy closure. Solution: Take two elements from

Let S be the set of all rational numbers of the form m/n, where m∈Z and n = 1,2,3. Prove that * on S defined by a*b = ab is not a binary operation. Read More »

Prove that the operation * on the set M = { [[a 0][0 b]] : a, b ∈ R – {0} } defined by A*B = AB is a binary operation.

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Binary Operation on Matrix Set 📺 Watch Video Explanation: Prove that the operation is a binary operation Given: Set \( M = \left\{ \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} : a, b \in \mathbb{R} – \{0\} \right\} \) Operation defined by: \( A * B = AB \) Concept: To prove a

Prove that the operation * on the set M = { [[a 0][0 b]] : a, b ∈ R – {0} } defined by A*B = AB is a binary operation. Read More »