Binary Operations — revision notes (CBSE Class 12)
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.
Binary Operations — Flashcards (CBSE Class 12)
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.
Binary Operations
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).
Forgetting closure. Students test commutativity and associativity but never confirm a ∗ b stays inside A. Example: subtraction is not a binary operation on ℕ because 3 − 5 = −2 ∉ ℕ. Also, commutativity and associativity are independent — one can hold without the other; never assume one from the other.
- ✓- A binary operation ∗ on A is a map A × A → A; closure is mandatory.
- ✓- Commutative: a ∗ b = b ∗ a. Associative: brackets don't matter.
- ✓- Identity e satisfies a ∗ e = e ∗ a = a and is unique if it exists.
- ✓- Inverse of a satisfies a ∗ a⁻¹ = e; needs an identity first.
- ✓- In a Cayley table: symmetry ⟹ commutative; a header-reproducing row ⟹ identity.
- ✓- Binary operations abstract "+, ×, min, HCF" into one framework.
- ✓- Always verify closure before anything else.
- ✓- Identity and inverses build toward groups — the language of higher algebra.
Binary Operations — Formula Sheet
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).
- ✓- Closure: result stays in A.
- ✓- Identity e satisfies a ∗ e = a.
- ✓- Inverse: a ∗ a⁻¹ = e.
A binary operation combines two elements to give a third; properties like identity and inverse require closure first.
Binary Operations — Worked Example
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.
- ✓- A binary operation combines two elements of a set to give another element of the same set.
- ✓- Commutative: a * b = b * a; Associative: (a * b) * c = a * (b * c).
- ✓- An identity element e satisfies a * e = e * a = a.