Commutativity and Associativity Check ๐บ Watch Video Explanation: Check commutativity and associativity Given: \( a * b = \frac{ab}{4}, \quad a,b \in \mathbb{Q} \) Commutativity: \( a * b = \frac{ab}{4} = \frac{ba}{4} = b * a \) โ Operation is commutative Associativity: LHS: \( (a*b)*c = \left(\frac{ab}{4}\right)*c = \frac{\frac{ab}{4} \cdot c}{4} = \frac{abc}{16} \) […]
Commutativity and Associativity Check ๐บ Watch Video Explanation: Check commutativity and associativity Given: \( a * b = a – b, \quad a,b \in \mathbb{Z} \) Commutativity: \( a * b = a – b \) \( b * a = b – a \) Example: \( 5 – 3 = 2 \neq 3 –
Commutativity and Associativity Check ๐บ Watch Video Explanation: Check commutativity and associativity Given: \( a * b = a^b, \quad a,b \in \mathbb{N} \) Commutativity: \( a * b = a^b \) \( b * a = b^a \) Example: \( 2^3 = 8 \neq 3^2 = 9 \) โ Operation is NOT commutative Associativity:
Commutativity and Associativity Check ๐บ Watch Video Explanation: Check commutativity and associativity Given: \( a * b = ab + 1, \quad a,b \in \mathbb{Q} \) Commutativity: \( a * b = ab + 1 \) \( b * a = ba + 1 = ab + 1 \) โ Operation is commutative Associativity: LHS:
Commutativity and Associativity Check ๐บ Watch Video Explanation: Check commutativity and associativity Given: \( a * b = (a – b)^2, \quad a,b \in \mathbb{Q} \) Commutativity: \( a * b = (a – b)^2 \) \( b * a = (b – a)^2 = (-(a – b))^2 = (a – b)^2 \) โ Operation
Commutativity and Associativity Check ๐บ Watch Video Explanation: Check commutativity and associativity Given: \( a * b = a + b – 7, \quad a,b \in \mathbb{R} \) Commutativity: \( a * b = a + b – 7 \) \( b * a = b + a – 7 = a + b –
Commutativity and Associativity Check ๐บ Watch Video Explanation: Check commutativity and associativity Given: \( a * b = a + ab = a(1 + b), \quad a,b \in \mathbb{Q} \) Commutativity: \( a * b = a + ab \) \( b * a = b + ba \) Clearly: \( a + ab \neq
Commutativity and Associativity Check ๐บ Watch Video Explanation: Check commutativity and associativity Given: \( a * b = ab^2, \quad a,b \in \mathbb{Q} \) Commutativity: \( a * b = ab^2 \) \( b * a = ba^2 \) Clearly: \( ab^2 \neq ba^2 \quad (\text{in general}) \) โ Operation is NOT commutative Associativity: LHS:
Commutativity and Associativity Check ๐บ Watch Video Explanation: Check commutativity and associativity Given: \( a \circ b = \frac{ab}{2}, \quad a,b \in \mathbb{Q} \) Commutativity: \( a \circ b = \frac{ab}{2} = \frac{ba}{2} = b \circ a \) โ Operation is commutative Associativity: LHS: \( (a \circ b)\circ c = \left(\frac{ab}{2}\right)\circ c = \frac{\frac{ab}{2} \cdot
Commutativity and Associativity Check ๐บ Watch Video Explanation: Check commutativity and associativity Given: \( a \circ b = a^2 + b^2, \quad a,b \in \mathbb{Q} \) Commutativity: \( a \circ b = a^2 + b^2 \) \( b \circ a = b^2 + a^2 = a^2 + b^2 \) โ Operation is commutative Associativity: LHS: