Understanding Binary Operations and Identity Elements

A binary operation is a rule that combines two elements from a set to produce another element from the same set. In this question, the binary operation is defined as \(a \square b = ab + (a + b)\), where \(a\) and \(b\) are any two real numbers. We are asked to find the identity element for this specific operation.

What is an Identity Element in Binary Operations?

An identity element, often denoted by \(x\) or \(e\), is a special element within a set that, when combined with any other element \(a\) from the set using a given binary operation, leaves \(a\) unchanged. For an operation \(\square\), an element \(x\) is an identity element if for every element \(a\) in the set:

• \(a \square x = a\) (right identity)
• \(x \square a = a\) (left identity)

If an element satisfies both conditions, it is called the identity element.

Finding the Identity Element for the Operation \(a \square b = ab + (a + b)\)

To find the identity element, let's denote it as \(x\). According to the definition of a right identity element, we must have:

\[a \square x = a\]

Now, we substitute the given definition of the binary operation \(a \square b = ab + (a + b)\) into our equation. Replacing \(b\) with \(x\), we get:

\[ax + (a + x) = a\]

Next, we simplify the left side of the equation by removing the parentheses:

\[ax + a + x = a\]

Our goal is to solve for \(x\). To do this, we can subtract \(a\) from both sides of the equation:

\[ax + a + x - a = a - a\]

This simplifies to:

\[ax + x = 0\]

Now, we can factor out \(x\) from the terms on the left side of the equation:

\[x(a + 1) = 0\]

For this equation \(x(a + 1) = 0\) to hold true for any real number \(a\), the term \(x\) must be equal to zero. Let's consider why:

• If \(a+1 \neq 0\) (i.e., \(a \neq -1\)), then for the product \(x(a+1)\) to be zero, \(x\) must be \(0\).
• If \(a+1 = 0\) (i.e., \(a = -1\)), then the equation becomes \(x(0) = 0\), which is true for any \(x\). However, the definition of an identity element requires it to work for *any* \(a\), not just specific values. Therefore, \(x\) must be \(0\) to satisfy the condition for all values of \(a\) (including when \(a \neq -1\)).

Verification of the Identity Element

Let's verify if \(x = 0\) indeed works as the identity element for the given binary operation \(a \square b = ab + (a + b)\):

Substitute \(x = 0\) back into the definition of the operation \(a \square x\):

\[a \square 0 = a(0) + (a + 0)\]

Simplify the expression:

\[a \square 0 = 0 + a\]

\[a \square 0 = a\]

Since \(a \square 0 = a\), this confirms that \(0\) is the identity element for the given binary operation.

The value of the identity element of this operation is \(0\).