Sets & Binary Operations

Sets & Binary Operations

Abstract algebra studies algebraic structures — sets equipped with operations that satisfy certain axioms. Before we dive into groups, we need to understand binary operations and their properties.

Binary Operations

A binary operation $*$ on a set $S$ is a function $*: S \times S \to S$. The operation is closed if for all $a, b \in S$, we have $a * b \in S$.

For example, addition on the integers $\mathbb{Z}$ is a binary operation: for any two integers $a$ and $b$, $a + b$ is also an integer.

Modular Arithmetic as a Set

The set $\mathbb{Z}_n = \{0, 1, 2, \ldots, n-1\}$ with addition modulo $n$ is a fundamental example. We can represent this in Python:

def mod_add(a, b, n):
    return (a + b) % n

def mod_mul(a, b, n):
    return (a * b) % n

Properties of Binary Operations

A binary operation $*$ on $S$ may have these properties:

• Associative: $(a * b) * c = a * (b * c)$ for all $a, b, c \in S$
• Commutative: $a * b = b * a$ for all $a, b \in S$
• Identity element: There exists $e \in S$ such that $e * a = a * e = a$ for all $a$
• Inverse: For each $a \in S$, there exists $a^{-1} \in S$ such that $a * a^{-1} = a^{-1} * a = e$

Cayley Table

A Cayley table displays the results of a binary operation. For $\mathbb{Z}_4$ under addition:

Your Task

Implement cayley_table(n) that returns the Cayley table (as a list of lists) for addition modulo $n$, and is_commutative(table) that checks whether a Cayley table represents a commutative operation.