Properties of a*b = a/b on Integers Question: Let \( * \) be defined on non-zero integers by: \[ a * b = \frac{a}{b} \] Which property is satisfied? (a) Closure (b) Commutative (c) Associative (d) None of these Solution: 1. Closure: For closure, result must be an integer. Example: \[ 1 * 2 = […]
Commutative Law a + b = b + a Question: The law \( a + b = b + a \) is called: (a) Closure law (b) Associative law (c) Commutative law (d) Distributive law Solution: The expression \[ a + b = b + a \] means that changing the order of operands does
Subtraction of Integers Properties Question: Subtraction of integers is: (a) Commutative but not associative (b) Commutative and associative (c) Associative but not commutative (d) Neither commutative nor associative Solution: Step 1: Check Commutativity \[ a – b \neq b – a \] Example: \[ 5 – 3 = 2 \neq -2 = 3 – 5
Check Commutativity & Associativity of a*b = ab + 1 Question: Let \( * \) be defined on \( \mathbb{R} \) by: \[ a * b = ab + 1 \] Determine its properties. Options: (a) Commutative but not associative (b) Associative but not commutative (c) Neither commutative nor associative (d) Both commutative and associative
Evaluate (2*3)*2 for a*b = a² + b² + ab + 1 Question: If \( a * b = a^2 + b^2 + ab + 1 \), find: \[ (2 * 3) * 2 \] Options: (a) 20 (b) 40 (c) 400 (d) 445 Solution: Step 1: Compute \( 2 * 3 \) \[ 2
Check Commutativity & Associativity of a*b = a + b + ab Question: The binary operation \( * \) defined on \( \mathbb{N} \) by: \[ a * b = a + b + ab \] Check its properties. Options: (a) Commutative only (b) Associative only (c) Commutative and associative both (d) None of these
Binary Operation MCQ on Z, Q, N Question: Which of the following statements is true? A. \( a * b = \frac{a+b}{2} \) is a binary operation on \( \mathbb{Z} \) B. \( a * b = \frac{a+b}{2} \) is a binary operation on \( \mathbb{Q} \) C. All commutative operations are associative D. Subtraction
Identity Element for a*b = a + b – ab Question: Let \( * \) be a binary operation on \( \mathbb{Q} – \{1\} \) defined by: \[ a * b = a + b – ab \] Find the identity element. Options: (a) 1 (b) \( \frac{a-1}{a} \) (c) \( \frac{a}{a-1} \) (d) 0
Evaluate 3 ⊙ (1/5 ⊙ 1/2) Question: If \( a ⊙ b = \frac{ab}{4} \), evaluate: \[ 3 ⊙ \left(\frac{1}{5} ⊙ \frac{1}{2}\right) \] Options: (a) \( \frac{3}{160} \) (b) \( \frac{5}{160} \) (c) \( \frac{3}{10} \) (d) \( \frac{3}{40} \) Solution: Step 1: Compute inner operation \[ \frac{1}{5} ⊙ \frac{1}{2} = \frac{\frac{1}{5} \cdot \frac{1}{2}}{4} =
Inverse of Element for a*b = ab/2 Question: Let \( \mathbb{Q}^+ \) be the set of all positive rational numbers. The binary operation \( * \) is defined by: \[ a * b = \frac{ab}{2} \] Find the inverse of an element \( a \in \mathbb{Q}^+ \). Options: (a) \( a \) (b) \( \frac{1}{a}