Educational

On the set Q of all rational numbers if a binary operation * is defined by a∗b=ab/5, prove that * is associative on Q.

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Associativity Proof 📺 Watch Video Explanation: Prove that the operation is associative Given: \( a * b = \frac{ab}{5}, \quad a,b \in \mathbb{Q} \) Proof: LHS: \( (a*b)*c = \left(\frac{ab}{5}\right)*c \) \( = \frac{\frac{ab}{5} \cdot c}{5} = \frac{abc}{25} \) RHS: \( a*(b*c) = a*\left(\frac{bc}{5}\right) \) \( = \frac{a \cdot \frac{bc}{5}}{5} = \frac{abc}{25} \) Thus: \( […]

On the set Q of all rational numbers if a binary operation * is defined by a∗b=ab/5, prove that * is associative on Q. Read More »

On Z, the set of all integers, a binary operation * is defined by a*b = a + 3b – 4. Prove that * is neither commutative nor associative on Z.

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Not Commutative and Not Associative 📺 Watch Video Explanation: Prove that the operation is neither commutative nor associative Given: \( a * b = a + 3b – 4, \quad a,b \in \mathbb{Z} \) 1. Not Commutative: Take \( a = 1 \), \( b = 2 \) \( a*b = 1 + 3(2) –

On Z, the set of all integers, a binary operation * is defined by a*b = a + 3b – 4. Prove that * is neither commutative nor associative on Z. Read More »

On Q, the set of all rational numbers, * is defined by a∗b = (a-b)/2 show that * is not associative.

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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 \( a = 2 \), \( b = 4 \), \( c = 6 \) LHS: \( (a*b)*c = \left(\frac{2-4}{2}\right)*6 = (-1)*6 \) \( =

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

Let S be the set of all real numbers except – 1 and let ‘*’ be an operation defined by a*b = a + b + ab for all a, b ∈S. Determine whether ‘*’ is a binary operation on ‘S’. if yes, Check its commutativity and associativity. Also, solve the equation (2*x)*3 = 7.

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Binary Operation Full Solution 📺 Watch Video Explanation: Given: \( a * b = a + b + ab, \quad a,b \in \mathbb{R} \setminus \{-1\} \) 1. Closure (Binary Operation Check): \( a*b = a + b + ab = (1+a)(1+b) – 1 \) Since \( a \neq -1 \) and \( b \neq -1

Let S be the set of all real numbers except – 1 and let ‘*’ be an operation defined by a*b = a + b + ab for all a, b ∈S. Determine whether ‘*’ is a binary operation on ‘S’. if yes, Check its commutativity and associativity. Also, solve the equation (2*x)*3 = 7. Read More »

On the set Z of integers a binary operation * is defined by a*b = ab + 1 for all a, b ∈Z. Prove that * is not associative on Z.

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

On the set Z of integers a binary operation * is defined by a*b = ab + 1 for all a, b ∈Z. Prove that * is not associative on Z. Read More »

If the binary operation ο is defined by aοb = a + b – ab on the set Q – { -1} of all rational numbers other than -1. Show that ο is commutative on Q – { – 1}.

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Commutativity Proof 📺 Watch Video Explanation: Show that the operation is commutative Given: \( a \circ b = a + b – ab, \quad a,b \in \mathbb{Q} \setminus \{-1\} \) Proof: Compute: \( a \circ b = a + b – ab \) \( b \circ a = b + a – ba \) Since:

If the binary operation ο is defined by aοb = a + b – ab on the set Q – { -1} of all rational numbers other than -1. Show that ο is commutative on Q – { – 1}. Read More »

Check the commutativity and associativity of the binary operations:‘*’ on Q defined by a*b = gcd(a, b) ∀ a, b ∈N

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GCD Binary Operation 📺 Watch Video Explanation: Check commutativity and associativity Given: \( a * b = \gcd(a,b), \quad a,b \in \mathbb{N} \) Commutativity: \( \gcd(a,b) = \gcd(b,a) \) ✔ Operation is commutative Associativity: \( \gcd(\gcd(a,b),c) = \gcd(a,\gcd(b,c)) \) ✔ Operation is associative Conclusion: ✔ The operation is both commutative and associative on \( \mathbb{N}

Check the commutativity and associativity of the binary operations:‘*’ on Q defined by a*b = gcd(a, b) ∀ 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: \( a * b = a + b – ab, \quad a,b \in \mathbb{Z} \) 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 »

Check the commutativity and associativity of the binary operations:‘*’ on Q defined by a*b = ab/4 ∀ a, b ∈Q

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Commutativity and Associativity Check 📺 Watch Video Explanation: Check commutativity and associativity Given: \( a * b = \frac{ab}{4}, \quad a,b \in \mathbb{Q} \) Commutativity: \( a * b = \frac{ab}{4} = \frac{ba}{4} = b * a \) ✔ Operation is commutative Associativity: LHS: \( (a*b)*c = \left(\frac{ab}{4}\right)*c = \frac{\frac{ab}{4} \cdot c}{4} = \frac{abc}{16} \)

Check the commutativity and associativity of the binary operations:‘*’ on Q defined by a*b = ab/4 ∀ a, b ∈Q Read More »