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 pairs of binary operations possess is distributivity, which means that applying one operation over the result of another is equivalent to applying the first operation to each operand individually and then combining the results. For example, for all real numbers $a$, $b$, and $c$, multiplication is distributive over addition: $a \times (b + c) = a \times b + a \times c$ (see also the examples below).

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

Algebra

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

\[a \cdot (b + c) = ab + ac\]

\[(b + c) \cdot a = ba + ca.\]

The distributive 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 ($\mathbb{N}$)

Let $a = 2$, $b = 3$, and $c = 4$.

$2 \cdot (3 + 4) = 2 \cdot 3 + 2 \cdot 4 = 6 + 8 = 14$
Integers ($\mathbb{Z}$)

Let $a = -2$, $b = 5$, and $c = -3$.

$-2 \cdot (5 + (-3)) = (-2) \cdot 5 + (-2) \cdot (-3) = -10 + 6 = -4$
Rational Numbers ($\mathbb{Q}$)

Let $a = \frac{1}{2}$, $b = \frac{1}{3}$, and $c = \frac{1}{4}$.

$\frac{1}{2} \cdot \left(\frac{1}{3} + \frac{1}{4}\right) = \frac{1}{2} \cdot \frac{1}{3} + \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{6} + \frac{1}{8} = \frac{4}{24} + \frac{3}{24} = \frac{7}{24}$
Real Numbers ($\mathbb{R}$)

Let $a = 1.2$, $b = -3.4$, and $c = 5.6$.

$ 1.2 \cdot ((-3.4) + 5.6) = 1.2 \cdot (-3.4) + 1.2 \cdot 5.6 = -4.08 + 6.72 = 2.64 $
Complex Numbers ($\mathbb{C}$)

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

$ (1 + 2i) \cdot ((3 + 4i) + (5 + 6i)) = (1 + 2i) \cdot (3 + 4i) + (1 + 2i) \cdot (5 + 6i) = (-5 + 10i) + (-7 + 16i) = -12 + 26i $

Set Theory

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

Examples

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

Union
$A \cup (B \cap C) = (A \cup B) \cap (A \cup C) = \{1, 2\}$
Intersection
$A \cap (B \cup C) = (A \cap B) \cup (A \cap C) = \{2\}$

Logical Equivalences

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

Examples

  • $p$: True (T).
  • $q$: False (F).
  • $r$: True (T).
Conjunction
T or (F and T) $=$ T, which is logically equivalent to: (T or F) and (T or T) $=$ T.
Disjunction
T and (F or T) = T, which is logically equivalent to: (T and F) or (T and 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

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