Binary Operations and Algebraic Structures: Notes and FAQs

A binary operation on a set S is a rule that combines any two elements of S to produce another element of S — the crucial requirement being closure: a ∗ b must land back inside S. Addition on the integers is a binary operation; subtraction on the natural numbers is not, because 2 − 5 = −3 escapes the set.

This humble definition is the foundation of all abstract algebra. Groups, rings, and fields are all just sets equipped with one or two binary operations satisfying extra rules — associativity, identity elements, inverses. Learning to test these properties on examples is the first real skill of the subject.

Key Concepts

• Binary operation: a function ∗: S × S → S; closure (a ∗ b ∈ S for all a, b) is part of the definition.
• Associativity: (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c — bracketing does not matter.
• Commutativity: a ∗ b = b ∗ a for all a, b — order does not matter (independent of associativity).
• Identity element: an element e with a ∗ e = e ∗ a = a for every a; if it exists, it is unique.
• Inverse: for an identity e, b is an inverse of a if a ∗ b = b ∗ a = e.
• Cayley table: the operation's multiplication table, useful for small finite sets.

How It Works

• Step 1 — closure: pick arbitrary a, b ∈ S and check a ∗ b ∈ S. Counterexample: division on ℤ fails since 1 ÷ 2 ∉ ℤ.
• Step 2 — associativity: expand both (a ∗ b) ∗ c and a ∗ (b ∗ c) algebraically and compare. Subtraction fails: (5 − 3) − 1 = 1 but 5 − (3 − 1) = 3.
• Step 3 — identity: solve a ∗ e = a for e in terms of a; the answer must be independent of a. For a ∗ b = a + b − ab on ℤ: a + e − ae = a gives e(1 − a) = 0 for all a, so e = 0.
• Step 4 — inverses: solve a ∗ b = e. For the same operation: a + b − ab = 0 gives b = a/(a − 1) for a ≠ 1 — not always an integer, so inverses fail on ℤ.
• Commutativity check: compare a ∗ b with b ∗ a; matrix multiplication is the classic non-commutative example.

Worked Example

Test the operation a ∗ b = ab (ordinary multiplication) on the set S = {0, 1}.

1. Closure: 0·0 = 0, 0·1 = 0, 1·0 = 0, 1·1 = 1 — all results are in {0, 1} ✓.
2. Associativity: inherited from integer multiplication: (a·b)·c = a·(b·c) ✓.
3. Commutativity: a·b = b·a ✓.
4. Identity: 1 works, since a·1 = 1·a = a for a ∈ {0, 1} ✓.
5. Inverses: 1 has inverse 1, but 0 has no inverse (nothing times 0 gives 1). So ({0, 1}, ·) has every property except universal inverses — it is a monoid, not a group.

Common Misconceptions

• "Binary just means two inputs." The outputs matter equally: without closure (results staying in S), it is not a binary operation on S at all.
• "Every operation has an identity." False — the operation a ∗ b = a + b + 1 on ℤ has identity −1 (check: a + (−1) + 1 = a), but a ∗ b = max(a, b) on ℤ has no identity.
• "Associativity is automatic." Subtraction, division, and exponentiation are all non-associative; associativity is a genuine restriction.

Real-World Applications

• Cryptography: modular addition and multiplication are binary operations whose group properties make RSA and Diffie–Hellman work.
• Computer science: string concatenation, boolean operations, and function composition are binary operations driving formal languages and logic circuits.
• Abstract algebra: every group, ring, and field starts life as a set with binary operations passing the tests above.

A binary operation is a closed combining rule, and the four property tests — closure, associativity, identity, inverses — sort operations into the algebraic structures built on them. Work through the steps mechanically on examples like a ∗ b = a + b − ab, and the definitions of semigroups, monoids, and groups will feel like natural next steps rather than new vocabulary.