Binary Operations

Binary operations generalise +, −, ×, ÷ into a single framework and are tested in CBSE as short 2–4 mark checks of properties on a given operation ∗. The definitions are quick to apply.

Definition

A binary operation ∗ on a set A is a function ∗ : A × A → A. It takes two elements of A and returns an element of A. The crucial requirement is closure: a ∗ b must lie in A for all a, b ∈ A.

Key properties

• Commutative: a ∗ b = b ∗ a for all a, b.
• Associative: (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c.
• Identity element e: a ∗ e = e ∗ a = a for all a. (At most one exists.)
• Inverse of a: an element b with a ∗ b = b ∗ a = e. (Needs identity to exist first.)

Worked example

On ℝ, define a ∗ b = a + b − ab.

• Commutative: a + b − ab = b + a − ba ✓
• Identity e: a + e − ae = a ⇒ e(1 − a) = 0 ⇒ e = 0. So identity is 0.
• Inverse of a: a + b − ab = 0 ⇒ b = a/(a − 1), valid for a ≠ 1.

Standard results

• On ℤ, subtraction is a binary operation but neither commutative nor associative.
• On ℕ, subtraction is NOT even a binary operation (not closed: 2 − 5 ∉ ℕ).

Exam Tricks & Tips

• 🎯 First always check closure — if a ∗ b can leave A, it's not a binary operation at all.
• 🎯 Identity is unique; solve a ∗ e = a to find it.
• 🎯 Find inverses only after confirming the identity exists.
• 🎯 A composition/operation table (Cayley table) makes commutativity visible: symmetry across the diagonal.
• 🎯 Subtraction and division fail associativity and commutativity — quick counter-examples score marks.
• ❌ Common mistake: hunting for inverses before establishing the identity element — inverse is defined relative to e.

Expected exam pattern

2–4 marks: verify whether a given ∗ is commutative/associative, find its identity and the inverse of an element.

Quick recap

Binary operation = closed map A × A → A. Test commutativity, associativity; find the unique identity e via a ∗ e = a, then inverses via a ∗ b = e.

Cover the answer, recall, then check. 11 cards on binary operations.

Q1. Define a binary operation on set A.
A1. A function ∗ : A × A → A; it must be closed (a ∗ b ∈ A always).

Q2. What is the closure property?
A2. For all a, b ∈ A, the result a ∗ b stays inside A.

Q3. State commutativity for ∗.
A3. a ∗ b = b ∗ a for all a, b ∈ A.

Q4. State associativity for ∗.
A4. (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c.

Q5. Define the identity element e.
A5. e with a ∗ e = e ∗ a = a for every a. It is unique if it exists.

Q6. Define the inverse of a.
A6. An element b with a ∗ b = b ∗ a = e (requires an identity e).

Q7. For a ∗ b = a + b − ab on ℝ, find the identity.
A7. e = 0, since a + 0 − 0 = a.

Q8. For that operation, find the inverse of a.
A8. b = a/(a − 1), valid for a ≠ 1.

Q9. Is subtraction on ℕ a binary operation?
A9. No — not closed (e.g. 2 − 5 = −3 ∉ ℕ).

Q10. Is subtraction on ℤ commutative or associative?
A10. Neither: 3 − 5 ≠ 5 − 3, and (8−3)−2 ≠ 8−(3−2).

Q11. How does a Cayley table reveal commutativity?
A11. The table is symmetric about its main diagonal.

Addition takes two numbers and returns one number. So does multiplication, and so does "the larger of a and b". Strip away the specifics and you get a binary operation — the abstract idea behind all of arithmetic, and the foundation of algebraic structures.

Definition: A binary operation ∗ on a set A is a function ∗ : A × A → A. It takes an ordered pair (a, b) of elements of A and returns a single element a ∗ b that must also lie in A. This last requirement is called closure and is built into the definition.

Properties a binary operation may have

• Commutative: a ∗ b = b ∗ a for all a, b ∈ A. (Addition and multiplication are; subtraction and division are not.)
• Associative: (a ∗ b) ∗ c = a ∗ (b ∗ c). (Addition, multiplication are; subtraction is not.)
• Identity element e: an element with a ∗ e = e ∗ a = a for all a. (0 for +, 1 for ×.) An identity, if it exists, is unique.
• Inverse of a: an element b with a ∗ b = b ∗ a = e. (−a for +, 1/a for × when a ≠ 0.) Inverses require an identity to exist first.

WHY these properties? They are the exact axioms that upgrade a bare set-with-operation into richer structures (groups, rings). Recognising them trains the algebraic thinking the whole syllabus rests on.

Reading an operation table

For a finite set, ∗ can be shown as a Cayley (composition) table. Quick diagnostics:

• Closure: every entry lies in the set.
• Commutativity: the table is symmetric about its main diagonal.
• Identity: some row (and matching column) reproduces the header exactly — that header element is e.

Worked example

Question: On ℚ (rationals), define a ∗ b = a + b − ab. Check closure, commutativity, associativity, find the identity, and find the inverse of a.

Solution:
Step 1 — Closure: sum and product of rationals are rational ⟹ a + b − ab ∈ ℚ. ✓
Step 2 — Commutative: a ∗ b = a + b − ab = b + a − ba = b ∗ a. ✓
Step 3 — Associative: (a ∗ b) ∗ c = (a+b−ab) + c − (a+b−ab)c = a + b + c − ab − bc − ca + abc. By symmetry a ∗ (b ∗ c) gives the same expression. ✓
Step 4 — Identity e: need a + e − ae = a ⟹ e(1 − a) = 0 for all a ⟹ e = 0. So identity is 0.
Step 5 — Inverse of a: need a + b − ab = 0 ⟹ b(1 − a) = −a ⟹ b = a/(a − 1), valid for a ≠ 1.
Conclusion: ∗ is closed, commutative, associative, with identity 0; every a ≠ 1 has inverse a/(a − 1).

Exam application

NCERT and boards love custom operations like a ∗ b = HCF(a,b), a ∗ b = min(a,b), or a ∗ b defined via a table, then ask you to test properties or find e and inverses. The moves are always the same four checks. Real-world echoes: matrix multiplication (associative, non-commutative), function composition (a binary operation on the set of functions), and modular arithmetic in cryptography.

Mnemonic — "CIA(C)": for a rich structure check Closure, Identity, Associativity, Commutativity (and inverses once identity is found).

Key formulas / definitions

• Binary operation ∗ on A: a map A × A → A (closure: a ∗ b ∈ A).
• Commutative: a ∗ b = b ∗ a.
• Associative: (a ∗ b) ∗ c = a ∗ (b ∗ c).
• Identity element e: a ∗ e = e ∗ a = a.
• Inverse of a: a ∗ b = b ∗ a = e (b = a⁻¹).

A binary operation combines two elements to give a third; properties like identity and inverse require closure first.

Worked Example

Problem: A binary operation * is defined on the set of real numbers by a * b = a + b − ab. (a) Compute 2 * 3. (b) Determine whether * is commutative.

Solution:
Step 1 — Apply the definition to compute 2 * 3:
2 * 3 = 2 + 3 − (2)(3) = 5 − 6 = −1.

Step 2 — Recall the meaning of commutativity: * is commutative if a * b = b * a for all a, b.

Step 3 — Compute a * b and b * a in general:
a * b = a + b − ab.
b * a = b + a − ba = a + b − ab (since addition and multiplication are commutative).

Step 4 — Compare. Both expressions are identical (a + b − ab), so a * b = b * a for all real a, b.

Answer: (a) 2 * 3 = −1; (b) the operation * is commutative.