Computation Rule and Properties

Definition: Binary Operation

A mapping \(\circ: X \times X \rightarrow X\) is a binary operation on \(X\) if for any pair of elements \((x_1, x_2)\) in \(X\), \(x_1 \circ x_2\) is also in \(X\).

Example: Addition (+) on the set of integers is a binary operation. For example, \(5 + 3 = 8\), and both \(5, 3, 8\) are integers, staying within the set of integers.

Definition: Associative Property

A binary operation \(\circ\) is associative if \((a \circ b) \circ c = a \circ (b \circ c)\) for all \(a, b, c \in X\).

Example: Multiplication of real numbers is associative: \((2 \times 3) \times 4 = 2 \times (3 \times 4) = 24\). In a modular context, we also have addition modulo \(n\) being associative. For example, for \(n = 5\), \((2 + 3) \bmod 5 + 4 \bmod 5 = 2 + (3 \bmod 5 + 4) \bmod 5 = 4\).

Definition: Commutative Property

A binary operation \(\circ\) is commutative if \(a \circ b = b \circ a\) for all \(a, b \in X\).

Example: Addition modulo \(n\) is also commutative. For \(n = 7\), \(5 + 3 \bmod 7 = 3 + 5 \bmod 7 = 1\).