set theory

Set theory notations

View version details

Set Theory

$\inline \displaystyle A,\, B,\, C,\, \ldots$			Sets containing elements of various types, e.g. $\inline \displaystyle A = \{\,$ red, green, blue $\inline \,\}$ , $\inline \displaystyle B = \{\,\pi,\, 2.71,\, -1.5\,\}$
$\inline \displaystyle \emptyset$			The empty set (the set containing no elements)
$\inline \displaystyle x \in A$			The element $\inline x$ belongs to the set $\inline A$ , e.g. $\inline \displaystyle \pi \in \{\,0,\, \pi,\, 2.71\,\}$ , red $\inline \displaystyle \in \{\,$ red, green, blue $\inline \displaystyle \,\}$
$\inline \displaystyle x \not\in A$			The element $\inline x$ does not belong to the set $\inline A$ , e.g. $\inline \displaystyle -1 \not\in \{\,0,\, \pi,\, 2.71\,\}$ , orange $\inline \displaystyle \not\in \{\,$ red, green, blue $\inline \displaystyle \,\}$
$\inline \displaystyle {\rm card}\, A$			The cardinality or the number of elements belonging to the set $\inline A$ , e.g. $\inline \displaystyle {\rm card}\, \{\,\pi,\, 2.71\,\} = 2$ , $\inline {\rm card}\, \{\,$ red, green, blue $\inline \displaystyle \,\} = 3$ , $\inline {\rm card}\, \emptyset = 0$ .
$\inline \displaystyle A \subseteq B$			The set $\inline A$ is contained inside the set $\inline B$ , thus all elements of $\inline A$ are also elements of $\inline B$ . In this situation we say that $\inline A$ is a subset of $\inline B$ .
$\inline \displaystyle A = B$			The sets $\inline A$ and $\inline B$ contain the same elements, e.g. $\inline \{\,$ a, b, c $\inline \,\} = \{\,$ b, a, c $\inline \,\} = \{\,$ c, b, a $\inline \,\}$ . The condition $\inline A = B$ is true if and only if $\inline A \subseteq B$ and $\inline B \subseteq A$ . When $\inline A$ does not contain the same elements as $\inline B$ the notation $\inline A \neq B$ is used.
$\inline \displaystyle A \subset B$			The set $\inline A$ is strictly contained inside the set $\inline B$ , in other words it is true that $\inline \displaystyle A \subseteq B$ but $\inline A \neq B$ .
$\inline \displaystyle \{\,x \mid P(x)\,\}$			The set of elements $\inline x$ satisfying the property $\inline P$ , e.g. for $\inline \displaystyle A = \{\,0,\, 1,\, 2,\, 3\,\}$ it follows that $\inline \displaystyle \{\,x \mid x \in A$ and $\inline \displaystyle x$ is even $\inline \displaystyle \,\} = \{\,0,\, 2\,\}$ .
$\inline \displaystyle \mathcal{P}(A)$			The power set of $\inline A$ is the set of all subsets of $\inline A$ , i.e. $\inline \displaystyle \mathcal{P}(A) = \{\,B \mid B \subseteq A\,\}$ . For example if $\inline \displaystyle A = \{\,0,\, 1\,\}$ then $\inline \displaystyle \mathcal{P}(A) = \{\,\emptyset,\, \{\,0\,\},\, \{\,1\,\},\, A\,\}$ .
$\inline \displaystyle A \cap B$			The intersection of sets $\inline A$ and $\inline B$ , i.e. the set containing the elements that are found both in $\inline A$ and $\inline B$ . Thus $\inline \displaystyle A \cap B = \{\,x \mid x \in A$ and $\inline \displaystyle x \in B\,\}$ . For example $\inline \displaystyle \{\,5,\, 1,\, 4,\, 2\,\} \cap \{\,0,\, 1,\, 2\,\} = \{\,1,\, 2\,\}$ .
$\inline \displaystyle \bigcap_{i=1}^n A_i$			The intersection of sets $\inline A_1$ , $\inline A_2$ , ..., $\inline A_n$ , i.e. the set containing the elements that are found in all of the given sets. In other words $\bigcap_{i=1}^n A_i = \{\,x \mid \forall i \in \{\,1,\, 2,\, \ldots,\, n\,\}:\, x \in A_i\,\}.$
$\inline \displaystyle A \cup B$			The union of sets $\inline A$ and $\inline B$ , i.e. the set containing the elements that are found in either $\inline A$ or $\inline B$ . Thus $\inline \displaystyle A \cup B = \{\,x \mid x \in A$ or $\inline \displaystyle x \in B\,\}$ . For example $\inline \displaystyle \{\,5,\, 1,\, 4,\, 2\,\} \cup \{\,0,\, 1,\, 2\,\} = \{\,0,\, 1,\, 2,\, 4,\, 5\,\}$ .
$\inline \displaystyle \bigcup_{i=1}^n A_i$			The union of sets $\inline A_1$ , $\inline A_2$ , ..., $\inline A_n$ , i.e. the set containing the elements that are found in at least one of the given sets. In other words $\bigcup_{i=1}^n A_i = \{\,x \mid \exists i \in \{\,1,\, 2,\, \ldots,\, n\,\}:\, x \in A_i \,\}.$
$\inline \displaystyle A \setminus B$			The difference between $\inline A$ and $\inline B$ , i.e. the set containing the elements that are found in $\inline A$ but are not found in $\inline B$ . Thus $\inline \displaystyle A \setminus B = \{\,x \mid x \in A$ and $\inline \displaystyle x \not\in B\,\}$ . For example $\inline \displaystyle \{\,5,\, 1,\, 4,\, 2\,\} \setminus \{\,0,\, 1,\, 2\,\} = \{\,4,\, 5\,\}$ .
$\inline \displaystyle A \times B$			The Cartesian product of $\inline A$ and $\inline B$ , i.e. the set of all possible ordered pairs whose first component is an element of $\inline A$ and whose second component is an element of $\inline B$ . Thus $\inline \displaystyle A \times B = \{\,(x,y) \mid x \in A$ and $\inline \displaystyle y \in B\,\}$ . For example $\inline \displaystyle \{\,0,\, 1\,\} \times \{\,a,\, b\,\} = \{\,(0,\, a),\, (0,\, b),\, (1,\, a),\, (1,\, b)\,\}$ .

$\inline \displaystyle \mathbb{N}$			The set of natural numbers, $\inline \displaystyle \mathbb{N} = \{\,0,\, 1,\, 2,\, 3,\, 4,\, \ldots \,\}$
$\inline \displaystyle \mathbb{N}^*$			The set of non-zero natural numbers, $\inline \displaystyle \mathbb{N}^* = \mathbb{N} \setminus \{\,0\,\} = \{\,1,\, 2,\, 3,\, 4,\, \ldots \,\}$
$\inline \displaystyle \mathbb{Z}$			The set of integers, $\inline \displaystyle \mathbb{Z} = \{\,\ldots,\, -2,\, -1,\, 0,\, 1,\, 2,\, \ldots\,\}$
$\inline \displaystyle \mathbb{Z}^*$			The set of non-zero integers, $\inline \displaystyle \mathbb{Z}^* = \mathbb{Z} \setminus \{\,0\,\} = \{\,\ldots,\, -2,\, -1,\, 1,\, 2,\, \ldots\,\}$
$\inline \displaystyle \mathbb{Z}_-$			The set of non-positive integers, $\inline \displaystyle \mathbb{Z}_- = \{\,\ldots,\, -4,\, -3,\, -2,\, -1,\, 0\,\}$
$\inline \displaystyle \mathbb{Z}_-^*$			The set of negative integers, $\inline \displaystyle \mathbb{Z}_-^* = \mathbb{Z}_- \setminus \{\,0\,\} = \{\,\ldots,\, -4,\, -3,\, -2,\, -1\,\}$
$\inline \displaystyle \mathbb{Q}$			The set of rational numbers, $\inline \displaystyle \mathbb{Q} = \left\{\, \left. \frac{a}{b} \,\,\right\|\, a \in \mathbb{Z},\, b \in \mathbb{Z}^*\,\right\}$
$\inline \displaystyle \mathbb{Q}^*$			The set of non-zero rational numbers, $\inline \displaystyle \mathbb{Q}^* = \mathbb{Q} \setminus \{\,0\,\} = \left\{\, \left. \frac{a}{b} \,\,\right\|\, a,\, b \in \mathbb{Z}^*\,\right\}$
$\inline \displaystyle \mathbb{Q}_-$			The set of non-positive rational numbers, $\inline \displaystyle \mathbb{Q}_- = \left\{\, \left.\frac{a}{b} \,\,\right\|\, a \in \mathbb{N},\, b \in \mathbb{Z}_-^*\,\right\}$
$\inline \displaystyle \mathbb{Q}_-^*$			The set of negative rational numbers, $\inline \displaystyle \mathbb{Q}_-^* = \mathbb{Q}_- \setminus \{\,0\,\} = \left\{\, \left. \frac{a}{b} \,\,\right\|\, a \in \mathbb{N}^,\, b \in \mathbb{Z}_-^\,\right\}$
$\inline \displaystyle \mathbb{Q}_+$			The set of non-negative rational numbers, $\inline \displaystyle \mathbb{Q}_+ = \left\{\, \left. \frac{a}{b} \,\,\right\|\, a \in \mathbb{N},\, b \in \mathbb{N}^*\,\right\}$
$\inline \displaystyle \mathbb{Q}_+^*$			The set of positive rational numbers, $\inline \displaystyle \mathbb{Q}_+^* = \mathbb{Q}_+ \setminus \{\,0\,\} = \left\{\, \left. \frac{a}{b} \,\,\right\|\, a,\, b \in \mathbb{N}^*\,\right\}$
$\inline \displaystyle \mathbb{R}$			The set of real numbers, $\inline \displaystyle \mathbb{R} = \mathbb{Q} \,\cup\, \{\,\sqrt{2},\, \sqrt{10},\, \pi,\, \mathrm{e},\, \ldots\,\}$ i.e. the union of the rationals and the irrationals
$\inline \displaystyle \mathbb{R} \setminus \mathbb{Q}$			The set of irrational numbers, $\inline \displaystyle \mathbb{R} \setminus \mathbb{Q} = \{\,\sqrt{2},\, \sqrt{10},\, \pi,\, \mathrm{e},\, \ldots\,\}$
$\inline \displaystyle \mathbb{R}^*$			The set of non-zero real numbers, $\inline \displaystyle \mathbb{R}^* = \mathbb{R} \setminus \{\,0\,\}$
$\inline \displaystyle \mathbb{R}_-$			The set of non-positive real numbers, $\inline \displaystyle \mathbb{R}_- = \{\,x \mid x \in \mathbb{R},\, x \leq 0\,\}$
$\inline \displaystyle \mathbb{R}_-^*$			The set of negative real numbers, $\inline \displaystyle \mathbb{R}_-^* = \mathbb{R}_- \setminus \{\,0\,\} = \{\,x \mid x \in \mathbb{R},\, x < 0\,\}$
$\inline \displaystyle \mathbb{R}_+$			The set of non-negative real numbers, $\inline \displaystyle \mathbb{R}_+ = \{\,x \mid x \in \mathbb{R},\, x \geq 0\,\}$
$\inline \displaystyle \mathbb{R}_+^*$			The set of positive real numbers, $\inline \displaystyle \mathbb{R}_+^* = \mathbb{R}_+ \setminus \{\,0\,\} = \{\,x \mid x \in \mathbb{R},\, x > 0\,\}$
$\inline \displaystyle [a,b],\, (a,b)$			Intervals on the real line defined through the sets: $[a,b] = \{\,x \in \mathbb{R} \,\mid\, a \leq x \leq b \,\} \f$ and \f$\displaystyle (a,b) = \{\,x \in \mathbb{R} \,\mid\, a < x < b \,\}$
$\inline \displaystyle (a,b],\, [a,b)$			Intervals on the real line defined through the sets: $(a,b] = \{\,x \in \mathbb{R} \,\mid\, a < x \leq b \,\} \f$ and \f$\displaystyle [a,b) = \{\,x \in \mathbb{R} \,\mid\, a \leq x < b \,\}$
$\inline \displaystyle (-\infty,a],\, (-\infty,a)$			Intervals on the real line defined through the sets: $(-\infty,a] = \{\,x \in \mathbb{R} \,\mid\, x \leq a \,\} \f$ and \f$\displaystyle (-\infty,a) = \{\,x \in \mathbb{R} \,\mid\, x < a \,\}$
$\inline \displaystyle [a,\infty),\, (a,\infty)$			Intervals on the real line defined through the sets: $[a, \infty) = \{\,x \in \mathbb{R} \,\mid\, x \geq a \,\} \f$ and \f$\displaystyle (a, \infty) = \{\,x \in \mathbb{R} \,\mid\, x > a \,\}$
$\inline \displaystyle \mathbb{C}$			The set of complex numbers, $\inline \displaystyle \mathbb{C} = \{\,a + b\,\mathrm{i} \mid a,\, b \in \mathbb{R}\,\}$ where $\inline \mathrm{i}$ is the imaginary unit satisfying $\inline \mathrm{i}^2 = -1$
$\inline \displaystyle \mathbb{C}^*$			The set of non-zero complex numbers, $\inline \displaystyle \mathbb{C}^* = \mathbb{C} \setminus \{\,0\,\}$
$\inline \displaystyle \mathbb{R}^n$			The $\inline n$ -dimensional real coordinate space, where $\inline n$ is a positive integer. This is basically the set containing all $\inline n$ -tuples of real numbers, defined by $\mathbb{R}^n = \{\, (x_1, x_2, \ldots, x_n) \,\mid\, x_i \in \mathbb{R},\, i = \overline{1,n} \,\}.$