Binary Operation

What it is

A binary operation $*$ on a set $S$ takes any two elements of $S$ and combines them into an element of $S$ (closure). An identity element $e$ leaves every element unchanged: $a*e = e*a = a$. An inverse of $a$ is an element $b$ with $a*b = b*a = e$.

Worked example: not a binary operation

Is subtraction a binary operation on the natural numbers? No: $2 - 5 = -3$ is not a natural number, so subtraction isn't closed on $\mathbb{N}$. It is a binary operation on the integers.

Worked example: identity and inverse

Find the identity element for $a*b = \dfrac{ab}{4}$ on the nonzero rationals, and the inverse of each element.

Identity: $a*e = a \Rightarrow \dfrac{ae}{4} = a \Rightarrow e = 4$, and $e*a = \dfrac{4a}{4} = a$ too.

Inverse: $a*b = 4 \Rightarrow \dfrac{ab}{4} = 4 \Rightarrow b = \dfrac{16}{a}$. For example, the inverse of 3 is $\dfrac{16}{3}$: $3 * \dfrac{16}{3} = \dfrac{16}{4} = 4$.

Worked example: commutative vs associative

For real numbers, $a*b = 1+ab$. Since $ab = ba$, it is commutative. For associativity:

These differ by $c - a$, so they agree only when $a = c$. The operation is commutative but not associative.

Counting binary operations

On a set with $n$ elements, a binary operation assigns one of $n$ results to each of the $n^2$ ordered pairs, so there are $n^{n^2}$ binary operations. With $n = 2$ that's $2^4 = 16$; with $n = 3$ it's $3^9 = 19683$.

Common mistakes

• Skipping closure. Check that every result lands back in the set before anything else.
• Checking identity on one side only. Both $a*e$ and $e*a$ must equal $a$ (they agree automatically only if the operation is commutative).
• Concluding associativity from a few numbers. One counterexample disproves it; proving it needs the general algebra.

Where this comes up in exams

Binary operations aren't listed in the 2025-26 CBSE Class XII syllabus, the JEE Main 2026 syllabus or the JEE Advanced 2026 syllabus, which were checked for this page. The ideas here (closure, identity, inverse, commutativity, associativity) are still useful background for relations, functions and matrices.

Checked against CBSE Mathematics curriculum, Classes XI-XII (2025-26), JEE (Main) 2026 syllabus (NTA) and JEE (Advanced) 2026 syllabus. Syllabi can change from year to year, so confirm with the latest official notification for your exam.

• Fit it into your week with the free timetable generator.
• Mark it off as you revise in the syllabus tracker.
• See how long you have left: exam countdowns.