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 $S$.

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

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

Algebra

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

\[a + b = b + a\] \[ab = ba.\]

The commutative 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 Multiplication
$3 + 5 = 5 + 3 = 8$ $3 \cdot 5 = 5 \cdot 3 = 15$
Integers ($\mathbb{Z}$)
Addition Multiplication
$(−3)+7=7+(−3)=4$ $(−2)×5=5×(−2)=−10$
Rational Numbers ($\mathbb{Q}$)
Addition Multiplication
$\frac{1}{2}+\frac{3}{4} = \frac{3}{4}+\frac{1}{2} = \frac{5}{4}$ $\frac{2}{3} \cdot \frac{3}{5} = \frac{3}{5} \cdot \frac{2}{3} = \frac{2}{5}$
Real Numbers ($\mathbb{R}$)
Addition Multiplication
$2.5 + 3.1 = 3.1 + 2.5 = 5.6$ $1.2 \cdot 4.5 = 4.5 \cdot 1.2 = 5.4$
Complex Numbers ($\mathbb{C}$)
Addition Multiplication
$(2 + 3i) + (1 - 4i) = (1 - 4i) + (2 + 3i) = 3 - i$ $(2 + 3i)(1 - 4i) = (1-4i)(2 + 3i) = 14 - 5i$

Set Theory

\[A \cup B = B \cup A\] \[A \cap B = B \cap A.\]

Examples

Let set $A = \{1, 2\}$ and set $B = \{1 + i, 2\}$.

Union Intersection
$A \cup B = B \cup A = \{1, 2, 1 + i\}$ $A \cap B = B \cap A = \{2\}$

Logical Equivalences

\[p \wedge q \equiv q \wedge p\] \[p \vee q \equiv q \vee p.\]

Examples

  • $p$: “The number $n$ is natural.”
  • $q$: “The number $n$ is even.”
Conjunction Disjunction
“The number $n$ is natural and even” is logically equivalent to “The number $n$ is even and natural.” “The number $n$ is natural or even” is logically equivalent to “The number $n$ is even or natural.”

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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