Identity Element for Special Matrix Set Question: Let \( G \) be the set of all matrices of the form: \[ \begin{bmatrix} x & x \\ x & x \end{bmatrix}, \quad x \in \mathbb{R} – \{0\} \] Find the identity element with respect to matrix multiplication. Solution: Step 1: Let identity matrix be \[ E […]
Inverse of 3 for a ⊙ b = ab/2 Question: Let \( \mathbb{Q}^+ \) be the set of positive rational numbers. If the binary operation \( \odot \) is defined by: \[ a \odot b = \frac{ab}{2} \] Find the inverse of 3. Options: (a) \( \frac{4}{3} \) (b) 2 (c) \( \frac{1}{3} \) (d)
Evaluate (2*3)*4 for a*b = 3a – b Question: If \( a * b = 3a – b \), find: \[ (2 * 3) * 4 \] Options: (a) 2 (b) 3 (c) 4 (d) 5 Solution: Step 1: Compute \( 2 * 3 \) \[ 2 * 3 = 3(2) – 3 = 6
Identity Element for a*b = a + b + 1 Question: For the binary operation \( * \) on \( \mathbb{Z} \) defined by: \[ a * b = a + b + 1 \] Find the identity element. Options: (a) 0 (b) -1 (c) 1 (d) 2 Solution: Step 1: Let identity be \(
Evaluate (2*3)*4 for a*b = a² – b² + ab + 4 Question: If \( a * b = a^2 – b^2 + ab + 4 \), find: \[ (2 * 3) * 4 \] Options: (a) 233 (b) 33 (c) 55 (d) -55 Solution: Step 1: Compute \( 2 * 3 \) \[ 2
Properties of Operation Δ on Power Set Question: On the power set \( P(A) \) of a non-empty set \( A \), define: \[ X \Delta Y = (X \cap Y) \cup (X \cap Y) \] Which of the following is true? (a) Commutative and associative without an identity (b) Commutative but not associative with
Evaluate 4 · 7 Using Max Operation Question: If \( a * b \) denotes the greater (maximum) of \( a \) and \( b \), and \( a \cdot b = (a * b) + 3 \), find: \[ 4 \cdot 7 \] Options: (a) 14 (b) 31 (c) 10 (d) 8 Solution: Step
Evaluate (4*5)*3 for a*b = a² + b² Question: If \( a * b = a^2 + b^2 \), find the value of: \[ (4 * 5) * 3 \] Options: (a) \( (4^2 + 5^2) + 3^2 \) (b) \( (4+5)^2 + 3^2 \) (c) \( 41^2 + 3^2 \) (d) \( (4 +
Find 3 * 4 for a*b = 2a + b – 3 Question: Let \( * \) be a binary operation on integers defined by: \[ a * b = 2a + b – 3 \] Find \( 3 * 4 \). Solution: Step 1: Substitute \( a = 3 \), \( b = 4
Find 22 * 4 Using HCF Question: Let \( * \) be a binary operation on \( \mathbb{N} \) defined by: \[ a * b = \text{HCF}(a, b) \] Find \( 22 * 4 \). Solution: Step 1: Apply definition \[ 22 * 4 = \text{HCF}(22, 4) \] Step 2: Find HCF Factors of 22: