Educational

Find 4 * 5 for a*b = 3a + 4b – 2 Question: Let \( * \) be a binary operation defined by: \[ a * b = 3a + 4b – 2 \] Find \( 4 * 5 \). Solution: Step 1: Substitute \( a = 4 \), \( b = 5 \) \[ […]

Evaluate 2 +₆ 4⁻¹ +₆ 3⁻¹ Question: Let \( +_6 \) be addition modulo 6 on the set \( S = \{0,1,2,3,4,5\} \). Evaluate: \[ 2 +_6 4^{-1} +_6 3^{-1} \] Concept: In addition modulo \(6\), the inverse of \(a\) is: \[ a^{-1} = 6 – a \] — Solution: Step 1: Find inverses \(4^{-1}

Identity Element for a*b = √(a² + b²) Question: A binary operation \( * \) is defined on \( \mathbb{R} \) by: \[ a * b = \sqrt{a^2 + b^2} \] Find the identity element. Concept: An identity element \( e \) satisfies: \[ a * e = a \quad \text{for all } a \in

Multiplication Modulo 5 Table on {0,1,2,3,4} Question: Write the composition table for the binary operation \( \times_5 \) (multiplication modulo 5) on the set \( S = \{0,1,2,3,4\} \). Concept: The operation is defined as: \[ a \times_5 b = (a \times b) \mod 5 \] Solution: Step 1: Multiply and take modulo 5 \(

Evaluate (3 ×₅ 4⁻¹)⁻¹ Modulo 5 Question: For the binary operation \( \times_5 \) on the set \( S = \{1,2,3,4\} \), evaluate: \[ (3 \times_5 4^{-1})^{-1} \] Solution: Step 1: Find inverse of 4 modulo 5 We need \( 4 \times x \equiv 1 \pmod{5} \) \(4 \times 4 = 16 \equiv 1 \)

Inverse of 3 Modulo 10 on {1,3,7,9} Question: For the binary operation \( \times_{10} \) on the set \( S = \{1,3,7,9\} \), write the inverse of 3. Concept: The inverse of an element \( a \) is an element \( b \in S \) such that: \[ a \times b \equiv 1 \pmod{10} \]

Multiplication Modulo 10 Table on {2,4,6,8} Question: Write the composition table for multiplication modulo 10 (\( \times_{10} \)) on the set \( S = \{2,4,6,8\} \). Concept: The operation is defined as: \[ a \times_{10} b = (a \times b) \mod 10 \] Solution: Step 1: Compute values \(2 \times 2 = 4 \equiv 4\)

Inverse of 5 Modulo 11 on {1,…,10} Question: Write the inverse of \(5\) under multiplication modulo \(11\) on the set \( \{1,2,\dots,10\} \). Concept: The inverse of \(a\) modulo 11 is a number \(b\) such that: \[ a \times b \equiv 1 \pmod{11} \] Solution: Step 1: Find \(b\) such that \[ 5 \times b

Solve 2*(x*5)=10 for a*b = ab/5 Question: Let \( * \) be a binary operation on non-zero real numbers defined by: \[ a * b = \frac{ab}{5} \] Find \( x \) if: \[ 2 * (x * 5) = 10 \] Solution: Step 1: Compute inner operation \[ x * 5 = \frac{x \cdot

Identity Element for a*b = 3ab/7 Question: Write the identity element for the binary operation \( * \) on the set \( \mathbb{R} \) defined by: \[ a * b = \frac{3ab}{7}, \quad \forall a,b \in \mathbb{R} \] Concept: An identity element \( e \) satisfies: \[ a * e = a \quad \text{and} \quad