Educational

Number of Binary Operations on Set with Two Elements Question: Write the total number of binary operations on a set consisting of two elements. Concept: If a set has \( n \) elements, then the number of binary operations on the set is: \[ n^{n^2} \] Solution: Given set has 2 elements, so \( n […]

Inverse of 4 under a*b = a + b + 2 Question: On the set \( \mathbb{Z} \) of all integers, a binary operation \( * \) is defined by: \[ a * b = a + b + 2 \] Find the inverse of 4. Solution: Step 1: Find Identity Element Let identity be

Identity Element for a*b = ab/2 Question: Write the identity element for the binary operation \( * \) on the set \( \mathbb{R}_0 \) (non-zero real numbers) defined by: \[ a * b = \frac{ab}{2}, \quad \forall a,b \in \mathbb{R}_0 \] Concept: An identity element \( e \) satisfies: \[ a * e = a

Identity and Inverse in Modulo 6 Addition Question: Define a binary operation \( * \) on the set \( S = \{0,1,2,3,4,5\} \) as: \[ a * b = \begin{cases} a + b, & \text{if } a + b < 6 \\ a + b - 6, & \text{if } a + b \geq 6

Binary Operation Table on Set {a,b,c,d} Question: Consider the binary operation \( \circ \) defined on the set \( S = \{a, b, c, d\} \) by the following table: \[ \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & a & a & a \\ b &

Binary Operation Table on Set {a,b,c,d} Question: Consider the binary operation \( * \) defined on the set \( S = \{a, b, c, d\} \) by the following table: \[ \begin{array}{c|cccc} * & a & b & c & d \\ \hline a & a & b & c & d \\ b &

Multiplication Table for Integers Modulo 5 Question: Write the multiplication table for the set of integers modulo 5, i.e., \( Z_5 = \{0,1,2,3,4\} \). Concept: In modulo 5 multiplication, \[ a \times_5 b = (a \times b) \mod 5 \] Each product is reduced to its remainder when divided by 5. Solution: Step 1: Compute

Inverse of 5 Modulo 11 Question: Find the inverse of \(5\) under multiplication modulo \(11\) on \(Z_{11}\). Concept: The inverse of an element \(a\) modulo \(11\) is a number \(b\) such that: \[ a \times b \equiv 1 \pmod{11} \] Solution: Step 1: Find \(b\) such that \[ 5 \times b \equiv 1 \pmod{11} \]

Compute 3⁻¹ ×₇ 4 (Modulo 7) Question: For the binary operation \( \times_7 \) on the set \( S = \{1,2,3,4,5,6\} \), compute \( 3^{-1} \times_7 4 \). Concept: The inverse of \( a \) under modulo 7 satisfies: \[ a \times a^{-1} \equiv 1 \pmod{7} \] Then compute the required operation using modulo 7.

Inverse of 3 under Multiplication Modulo 10 Question: For the binary operation \( \times_{10} \) on the set \( S = \{1,3,7,9\} \), find the inverse of 3. Concept: The inverse of an element \( a \) under multiplication modulo 10 is an element \( b \in S \) such that: \[ a \times_{10} b