Associativity

Definition

Let $S$ be a set and let $\ast$ be a binary operation on $S$. The operation $\ast$ is associative when $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a, b, c \in S$.

Remarks

Addition and multiplication of real numbers are associative. Subtraction of integers is not associative: $(1 - 1) - 1 = -1$, but $1 - (1 - 1) = 1$. Composition of maps from a set to itself is associative. For an associative operation, each way to put parentheses in $a_1 \ast a_2 \ast \cdots \ast a_n$ gives the same result.

Explanations

L1 human

Associativity says that you can do the operations in a chain in any grouping. For example, $(2 + 3) + 4$ and $2 + (3 + 4)$ are both $9$. Thus we can write $2 + 3 + 4$ without parentheses.