Binary Operations: Closure, Commutative, Associative and Distributive Laws, Identity and Inverse

What is a binary operation?

A binary operation on a set S is a rule that takes two elements of S and combines them to give one result. Addition, subtraction and multiplication of numbers are binary operations. In exams, new operations are defined with symbols such as ∗, ∘, ⊕ or Δ.

The properties (laws) at a glance

• Closure: a ∗ b is always in S. 🔒
• Commutative: a ∗ b = b ∗ a. The order changes. 🔄 Swap
• Associative: (a ∗ b) ∗ c = a ∗ (b ∗ c). The grouping changes, never the order. 🧩 Group
• Distributive (∗ over +): a ∗ (b + c) = (a ∗ b) + (a ∗ c). The outside element spreads over the bracket. 🎯 Spread
• Identity element e: a ∗ e = e ∗ a = a for every a.
• Inverse a⁻¹: a ∗ a⁻¹ = a⁻¹ ∗ a = e.

Closure

S is closed under ∗ if combining any two elements of S always gives an element of S.

• The whole numbers are closed under + and ×, but not under − (3 − 5 = −2 is not a whole number).
• The integers are not closed under ÷ (1 ÷ 2 is not an integer).

Commutative law

Associative law

Subtraction is a classic trap: (8 − 5) − 2 = 1 but 8 − (5 − 2) = 5, so subtraction is not associative (and not commutative either).

Distributive law

Ordinary multiplication is distributive over addition: 3(4 + 5) = 3(4) + 3(5) = 27.

Finding the identity element

Solve a ∗ e = a for e. The identity must be a single number that does not depend on a.

Finding inverses

Solve a ∗ a⁻¹ = e, using the identity you have just found.

Operation (Cayley) tables

A finite set can be shown in a table: the entry in row a and column b is a ∗ b.

How to read a table:

• Closed? Every entry is a member of the set.
• Commutative? The table is a mirror image across the leading diagonal (top-left to bottom-right).
• Identity? Find the row that copies the header row exactly: the element at the start of that row is e.
• Inverses? Find where e appears in each row; the column heading is that element's inverse.

Common mistakes

1. Working left to right instead of doing the bracket first.
2. "Proving" a law with one example: use letters to prove, one counter-example to disprove.
3. Giving an identity that contains a, such as e = a/(1 + a): that is not an identity.
4. Forgetting to state the value(s) with no inverse (where the denominator is zero).
5. Moving the order in the associative test: only the brackets move.

Practice

1. a ∗ b = 3a − 2b. Find (4 ∗ 2) ∗ 1.
2. Is a ∗ b = a + b + 5 commutative? Is it associative?
3. Find the identity element and the inverse of a for a ∗ b = a + b + 5.
4. For a ∗ b = 2ab on the non-zero real numbers, find the identity and the inverse of 3.
5. Is a ∗ b = a − b + ab commutative? Give a reason.
6. In multiplication modulo 5, solve 2 ⊗ x = 4.

Answers: 1. 4 ∗ 2 = 12 − 4 = 8, then 8 ∗ 1 = 24 − 2 = 22 · 2. Yes and yes: (a ∗ b) ∗ c = a ∗ (b ∗ c) = a + b + c + 10 · 3. a + e + 5 = a, so e = −5; a + a⁻¹ + 5 = −5, so a⁻¹ = −10 − a · 4. 2ae = a gives e = ½; 2(3)x = ½ gives x = 1/12 · 5. No: 2 ∗ 1 = 2 − 1 + 2 = 3 but 1 ∗ 2 = 1 − 2 + 2 = 1 · 6. x = 2 (2 × 2 = 4)

Next class: Mathematical Induction.