A binary operation * is defined as the set,R,of a real number by m*n=m+n+2.find the identity element

To find the identity element for the binary operation \( * \) defined on the set \( \mathbb{R} \) of real numbers by \( m * n = m + n + 2 \), we need to find an element \( e \in \mathbb{R} \) such that for any \( m \in \mathbb{R} \):

\[
m * e = m \quad \text{and} \quad e * m = m
\]

Given the operation:

\[
m * e = m + e + 2
\]

We set this equal to \( m \) to find \( e \):

\[
m + e + 2 = m
\]

Subtract \( m \) from both sides:

\[
e + 2 = 0
\]

Solving for \( e \):

\[
e = -2
\]

Thus, the identity element \( e \) is \(-2\).

**Would you like more details or have any questions?**

Here are 8 related questions that expand on this topic:

1. What is the inverse element of a given number under this operation?
2. Can you prove that the operation is associative?
3. What would happen if the operation was defined as \( m * n = m + n + c \) with a different constant \( c \)?
4. How does this binary operation differ from the standard addition operation on real numbers?
5. Can you find the identity element if the operation was \( m * n = m + n - 2 \)?
6. Is the operation commutative, and how would you prove it?
7. What is the result of \( 3 * 5 \) under this operation?
8. How would you define a similar operation on complex numbers?

**Tip:** When working with binary operations, always check for properties like associativity, commutativity, and the existence of identity and inverse elements.

Discover how to find the identity element for the binary operation * defined on the set of real numbers. Explore step-by-step solutions and related questions on binary operations and identity elements in mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation