Associative Laws

Taken and adapted from:

- Tom M. Apostol, Calculus, Volume 1 (New York: Blaisdell, 1961), pp. 13-19.
- Susanna S. Epp, Discrete mathematics: with applications, fifth edition, metric version (Australia, 2020), Appendix A.

Many fundamental operations in mathematics involve combining two elements from a set to produce another element of the same set; such rules are known as binary operations. For instance, addition and multiplication are binary operations on the real numbers, because adding or multiplying any two real numbers always yields another real number. Formally, if $ * $ is a binary operation on $S$, then $* : S \times S \to S$, meaning for any $a, b \in S$, the result $a * b$ is also in $S$. This property is called closure under the operation.

An important property that some binary operations possess is associativity, which means that the grouping of the operands does not affect the result. For example, for all real numbers $a$, $b$, and $c$, both addition and multiplication satisfy $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$ (see also the examples below).

This text also presents how similar associative properties apply to certain set and logical operations, as shown below.

Algebra

For all numbers $a$, $b$, and $c$ $(\forall \; a, b, c \in \mathbb{C})$,

\[(a + b) + c = a + (b + c)\]

\[(ab)c = a(bc).\]

The associative laws are valid within the following sets of numbers:

Natural Numbers: $\mathbb{N},$
Integers: $\mathbb{Z}$,
Rational Numbers: $\mathbb{Q}$,
Real Numbers: $\mathbb{R}$,
Complex Numbers: $\mathbb{C}.$

Examples

Natural Numbers ($\mathbb{N}$)

Addition
$(2 + 3) + 4 = 2 + (3 + 4) = 9$

Multiplication
$(2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4) = 24$

Integers ($\mathbb{Z}$)

Addition
$((-2) + 5) + (-3) = -2 + (5 + (-3)) = 0$

Multiplication
$((-2) \cdot 5) \cdot (-3) = (-2) \cdot (5 \cdot -3) = 30$

Rational Numbers ($\mathbb{Q}$)

Addition
$\left(\frac{1}{2}+\frac{1}{3}\right)+\frac{1}{4}=\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right)=\frac{13}{12}$

Multiplication
$\left(\frac{1}{2}\cdot\frac{1}{3}\right)\cdot\frac{1}{4}=\frac{1}{2}\cdot\left(\frac{1}{3}\cdot\frac{1}{4}\right)=\frac{1}{24}$

Real Numbers ($\mathbb{R}$)

Addition
$(1.2 + (-3.4)) + 5.6 = 1.2 + ((-3.4) + 5.6) = 3.4$

Multiplication
$(1.2\cdot(-3.4))\cdot 5.6=1.2\cdot((-3.4)\cdot 5.6)=-22.848$

Complex Numbers ($\mathbb{C}$)

Let $a = 1 + 2i$, $b = 3 + 4i$, and $c = 5 + 6i$

Addition
$(a + b) + c = a + (b + c) = 9 + 12i$

Multiplication
$(a \cdot b) \cdot c = a \cdot (b \cdot c) = -85 + 20i$

Set Theory

\[(A \cup B) \cup C = A \cup (B \cup C)\] \[(A \cap B) \cap C = A \cap (B \cap C).\]

Examples

Let set $A = \{1, 2\}$, set $B = \{2, 3\}$, and set $C=\{2, 4\}$.

Union
$(A \cup B) \cup C = A \cup (B \cup C) = \{1, 2, 3, 4\}$

Intersection
$(A \cap B) \cap C = A \cap (B \cap C) = \{2\}$

Logical Equivalences

\[(p \wedge q) \wedge r \equiv p \wedge (q \wedge r)\] \[(p \vee q) \vee r \equiv p \vee (q \vee r).\]

Examples

$p$: True (T).
$q$: False (F).
$r$: True (T).

Conjunction
(T and F) and T $=$ F and T $=$ F, which is logically equivalent to: T and (F and T) $=$ T and F $=$ F.

Disjunction
(T or F) or T $=$ T or T $=$ T, which is logically equivalent to: T or (F or T) $=$ T or T $=$ T.

Meaning of the used symbols

Symbol	Meaning
$ \forall $	for all (universal quantifier)
$ \in $	is an element of
$ \cup $	union of sets
$ \cap $	intersection of sets
$ \wedge $	and
$ \equiv $	is logically equivalent to
$ \vee $	(not exclusive) or