If the binary operation $\ast$ is defined by m $\ast$ n = mn + m + n for any real number m and n, find the identity of the elements under this operation

Analysis

To find the identity element ($e$) of a binary operation, we use the fundamental property: $m \ast e = m$ (or $e \ast m = m$) This means that when you perform the operation between any element $m$ and the identity $e$, the result must be the element $m$ itself.

Concept

The identity element is the "neutral" element of an operation. Just as $0$ is the identity for addition ($m + 0 = m$) and $1$ is the identity for multiplication ($m \times 1 = m$), we need to find the value of $e$ that satisfies the given rule: $m \ast n = mn + m + n$

Step-by-Step Solution

1. Set up the equation: Using the definition $m \ast e = m$, substitute $e$ for $n$ in the given formula: $m \ast e = me + m + e$

2. Equate to $m$: Since $m \ast e$ must equal $m$: $me + m + e = m$

3. Solve for $e$: Subtract $m$ from both sides of the equation: $me + e = 0$

4. Factor out $e$: $e(m + 1) = 0$

5. Determine the value of $e$: For this equation to hold true for any real number $m$, the term $(m + 1)$ cannot be zero (unless $m = -1$). Therefore, the only way for the product to be zero for all $m$ is if $e = 0$.

Verification

Let's check if $e = 0$ works: $m \ast 0 = (m \times 0) + m + 0$ $m \ast 0 = 0 + m + 0$ $m \ast 0 = m$ Since the result is $m$, $e = 0$ is indeed the identity element.

Conclusion

The correct option is D.

Exam Tip: Whenever you are asked to find the identity element, always remember the rule $a \ast e = a$. If you are asked for the inverse element later, remember that $a \ast a^{-1} = e$ (where $e$ is the identity you just found).