Binary operation

A binary operation is a calculation that combines two elements (called operands) to produce a third element. It takes two inputs from a set and produces a single output that is also part of that set.

Key characteristics:

• Inputs: Always takes exactly two inputs.
• Output: Produces exactly one output.
• Closure: For an operation to be binary on a set, the result of the operation on any two elements of the set must also be an element of the same set.

Common examples of binary operations include:

• Addition ($+$) on real numbers: $a + b$
• Subtraction ($-$) on real numbers: $a - b$
• Multiplication ($\times$) on real numbers: $a \times b$
• Division ($\div$) on non-zero real numbers: $a \div b$

Let's consider an example using a custom binary operation.
Suppose we define a binary operation $\ast$ on the set of integers $\mathbb{Z}$ as $a \ast b = a^2 + b$.

Example: Calculate $3 \ast 5$.