set theory
Set theory notations
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Set Theory
| $ \displaystyle A,\, B,\, C,\, \ldots$ | Sets containing elements of various types, e.g. $\displaystyle A = \{\,$ red, green, blue $\,\}$, $\displaystyle B = \{\,\pi,\, 2.71,\, -1.5\,\}$ | ||
| $ \displaystyle \emptyset$ | The empty set (the set containing no elements) | ||
| $\displaystyle x \in A$ | The element $x$ belongs to the set $A$, e.g. $\displaystyle \pi \in \{\,0,\, \pi,\, 2.71\,\}$, red $\displaystyle \in \{\,$ red, green, blue $\displaystyle \,\}$ | ||
| $ \displaystyle x \not\in A$ | The element $x$ does not belong to the set $A$, e.g. $\displaystyle -1 \not\in \{\,0,\, \pi,\, 2.71\,\}$, orange $\displaystyle \not\in \{\,$ red, green, blue $\displaystyle \,\}$ | ||
| $ \displaystyle {\rm card}\, A$ | The cardinality or the number of elements belonging to the set $A$, e.g. $\displaystyle {\rm card}\, \{\,\pi,\, 2.71\,\} = 2$, ${\rm card}\, \{\,$ red, green, blue $\displaystyle \,\} = 3$, ${\rm card}\, \emptyset = 0$. | ||
| $ \displaystyle A \subseteq B$ | The set $A$ is contained inside the set $B$, thus all elements of $A$ are also elements of $B$. In this situation we say that $A$ is a subset of $B$. | ||
| $ \displaystyle A = B$ | The sets $A$ and $B$ contain the same elements, e.g. $\{\,$a, b, c$\,\} = \{\,$b, a, c$\,\} = \{\,$c, b, a$\,\}$. The condition $A = B$ is true if and only if $A \subseteq B$ and $B \subseteq A$. When $A$ does not contain the same elements as $B$ the notation $A \neq B$ is used. | ||
| $ \displaystyle A \subset B$ | The set $A$ is strictly contained inside the set $B$, in other words it is true that $\displaystyle A \subseteq B$ but $A \neq B$. | ||
| $ \displaystyle \{\,x \mid P(x)\,\}$ | The set of elements $x$ satisfying the property $P$, e.g. for $\displaystyle A = \{\,0,\, 1,\, 2,\, 3\,\}$ it follows that $\displaystyle \{\,x \mid x \in A$ and $\displaystyle x$ is even $\displaystyle \,\} = \{\,0,\, 2\,\}$. | ||
| $ \displaystyle \mathcal{P}(A)$ | The power set of $A$ is the set of all subsets of $A$, i.e. $\displaystyle \mathcal{P}(A) = \{\,B \mid B \subseteq A\,\}$. For example if $\displaystyle A = \{\,0,\, 1\,\}$ then $\displaystyle \mathcal{P}(A) = \{\,\emptyset,\, \{\,0\,\},\, \{\,1\,\},\, A\,\}$. | ||
| $ \displaystyle A \cap B$ | The intersection of sets $A$ and $B$, i.e. the set containing the elements that are found both in $A$ and $B$. Thus $\displaystyle A \cap B = \{\,x \mid x \in A$ and $\displaystyle x \in B\,\}$. For example $\displaystyle \{\,5,\, 1,\, 4,\, 2\,\} \cap \{\,0,\, 1,\, 2\,\} = \{\,1,\, 2\,\}$. | ||
| $ \displaystyle \bigcap_{i=1}^n A_i$ | The intersection of sets $A_1$, $A_2$, ..., $A_n$, i.e. the set containing the elements that are found in all of the given sets. In other words \( \bigcap_{i=1}^n A_i = \{\,x \mid \forall i \in \{\,1,\, 2,\, \ldots,\, n\,\}:\, x \in A_i\,\}. \) | ||
| $ \displaystyle A \cup B$ | The union of sets $A$ and $B$, i.e. the set containing the elements that are found in either $A$ or $B$. Thus $\displaystyle A \cup B = \{\,x \mid x \in A$ or $\displaystyle x \in B\,\}$. For example $\displaystyle \{\,5,\, 1,\, 4,\, 2\,\} \cup \{\,0,\, 1,\, 2\,\} = \{\,0,\, 1,\, 2,\, 4,\, 5\,\}$. | ||
| $ \displaystyle \bigcup_{i=1}^n A_i$ | The union of sets $A_1$, $A_2$, ..., $A_n$, i.e. the set containing the elements that are found in at least one of the given sets. In other words \( \bigcup_{i=1}^n A_i = \{\,x \mid \exists i \in \{\,1,\, 2,\, \ldots,\, n\,\}:\, x \in A_i \,\}. \) | ||
| $ \displaystyle A \setminus B$ | The difference between $A$ and $B$, i.e. the set containing the elements that are found in $A$ but are not found in $B$. Thus $\displaystyle A \setminus B = \{\,x \mid x \in A$ and $\displaystyle x \not\in B\,\}$. For example $\displaystyle \{\,5,\, 1,\, 4,\, 2\,\} \setminus \{\,0,\, 1,\, 2\,\} = \{\,4,\, 5\,\}$. | ||
| $ \displaystyle A \times B$ | The Cartesian product of $A$ and $B$, i.e. the set of all possible ordered pairs whose first component is an element of $A$ and whose second component is an element of $B$. Thus $\displaystyle A \times B = \{\,(x,y) \mid x \in A$ and $\displaystyle y \in B\,\}$. For example $\displaystyle \{\,0,\, 1\,\} \times \{\,a,\, b\,\} = \{\,(0,\, a),\, (0,\, b),\, (1,\, a),\, (1,\, b)\,\}$. |
| $ \displaystyle \mathbb{N}$ | The set of natural numbers, $\displaystyle \mathbb{N} = \{\,0,\, 1,\, 2,\, 3,\, 4,\, \ldots \,\}$ | ||
| $ \displaystyle \mathbb{N}^*$ | The set of non-zero natural numbers, $\displaystyle \mathbb{N}^* = \mathbb{N} \setminus \{\,0\,\} = \{\,1,\, 2,\, 3,\, 4,\, \ldots \,\}$ | ||
| $ \displaystyle \mathbb{Z}$ | The set of integers, $\displaystyle \mathbb{Z} = \{\,\ldots,\, -2,\, -1,\, 0,\, 1,\, 2,\, \ldots\,\}$ | ||
| $ \displaystyle \mathbb{Z}^*$ | The set of non-zero integers, $\displaystyle \mathbb{Z}^* = \mathbb{Z} \setminus \{\,0\,\} = \{\,\ldots,\, -2,\, -1,\, 1,\, 2,\, \ldots\,\}$ | ||
| $ \displaystyle \mathbb{Z}_-$ | The set of non-positive integers, $\displaystyle \mathbb{Z}_- = \{\,\ldots,\, -4,\, -3,\, -2,\, -1,\, 0\,\}$ | ||
| $ \displaystyle \mathbb{Z}_-^*$ | The set of negative integers, $\displaystyle \mathbb{Z}_-^* = \mathbb{Z}_- \setminus \{\,0\,\} = \{\,\ldots,\, -4,\, -3,\, -2,\, -1\,\}$ | ||
| $ \displaystyle \mathbb{Q}$ | The set of rational numbers, $\displaystyle \mathbb{Q} = \left\{\, \left. \frac{a}{b} \,\,\right|\, a \in \mathbb{Z},\, b \in \mathbb{Z}^*\,\right\}$ | ||
| $ \displaystyle \mathbb{Q}^*$ | The set of non-zero rational numbers, $\displaystyle \mathbb{Q}^* = \mathbb{Q} \setminus \{\,0\,\} = \left\{\, \left. \frac{a}{b} \,\,\right|\, a,\, b \in \mathbb{Z}^*\,\right\}$ | ||
| $ \displaystyle \mathbb{Q}_-$ | The set of non-positive rational numbers, $\displaystyle \mathbb{Q}_- = \left\{\, \left.\frac{a}{b} \,\,\right|\, a \in \mathbb{N},\, b \in \mathbb{Z}_-^*\,\right\}$ | ||
| $ \displaystyle \mathbb{Q}_-^*$ | The set of negative rational numbers, $\displaystyle \mathbb{Q}_-^* = \mathbb{Q}_- \setminus \{\,0\,\} = \left\{\, \left. \frac{a}{b} \,\,\right|\, a \in \mathbb{N}^*,\, b \in \mathbb{Z}_-^*\,\right\}$ | ||
| $ \displaystyle \mathbb{Q}_+$ | The set of non-negative rational numbers, $\displaystyle \mathbb{Q}_+ = \left\{\, \left. \frac{a}{b} \,\,\right|\, a \in \mathbb{N},\, b \in \mathbb{N}^*\,\right\}$ | ||
| $ \displaystyle \mathbb{Q}_+^*$ | The set of positive rational numbers, $\displaystyle \mathbb{Q}_+^* = \mathbb{Q}_+ \setminus \{\,0\,\} = \left\{\, \left. \frac{a}{b} \,\,\right|\, a,\, b \in \mathbb{N}^*\,\right\}$ | ||
| $ \displaystyle \mathbb{R}$ | The set of real numbers, $\displaystyle \mathbb{R} = \mathbb{Q} \,\cup\, \{\,\sqrt{2},\, \sqrt{10},\, \pi,\, \mathrm{e},\, \ldots\,\}$ i.e. the union of the rationals and the irrationals | ||
| $ \displaystyle \mathbb{R} \setminus \mathbb{Q}$ | The set of irrational numbers, $\displaystyle \mathbb{R} \setminus \mathbb{Q} = \{\,\sqrt{2},\, \sqrt{10},\, \pi,\, \mathrm{e},\, \ldots\,\}$ | ||
| $ \displaystyle \mathbb{R}^*$ | The set of non-zero real numbers, $\displaystyle \mathbb{R}^* = \mathbb{R} \setminus \{\,0\,\}$ | ||
| $ \displaystyle \mathbb{R}_-$ | The set of non-positive real numbers, $\displaystyle \mathbb{R}_- = \{\,x \mid x \in \mathbb{R},\, x \leq 0\,\}$ | ||
| $ \displaystyle \mathbb{R}_-^*$ | The set of negative real numbers, $\displaystyle \mathbb{R}_-^* = \mathbb{R}_- \setminus \{\,0\,\} = \{\,x \mid x \in \mathbb{R},\, x < 0\,\}$ | ||
| $ \displaystyle \mathbb{R}_+$ | The set of non-negative real numbers, $\displaystyle \mathbb{R}_+ = \{\,x \mid x \in \mathbb{R},\, x \geq 0\,\}$ | ||
| $ \displaystyle \mathbb{R}_+^*$ | The set of positive real numbers, $\displaystyle \mathbb{R}_+^* = \mathbb{R}_+ \setminus \{\,0\,\} = \{\,x \mid x \in \mathbb{R},\, x > 0\,\}$ | ||
| $ \displaystyle [a,b],\, (a,b)$ | Intervals on the real line defined through the sets: \( [a,b] = \{\,x \in \mathbb{R} \,\mid\, a \leq x \leq b \,\} \f$ and \f$\displaystyle (a,b) = \{\,x \in \mathbb{R} \,\mid\, a < x < b \,\} \) | ||
| $ \displaystyle (a,b],\, [a,b)$ | Intervals on the real line defined through the sets: \( (a,b] = \{\,x \in \mathbb{R} \,\mid\, a < x \leq b \,\} \f$ and \f$\displaystyle [a,b) = \{\,x \in \mathbb{R} \,\mid\, a \leq x < b \,\} \) | ||
| $ \displaystyle (-\infty,a],\, (-\infty,a)$ | Intervals on the real line defined through the sets: \( (-\infty,a] = \{\,x \in \mathbb{R} \,\mid\, x \leq a \,\} \f$ and \f$\displaystyle (-\infty,a) = \{\,x \in \mathbb{R} \,\mid\, x < a \,\} \) | ||
| $ \displaystyle [a,\infty),\, (a,\infty)$ | Intervals on the real line defined through the sets: \( [a, \infty) = \{\,x \in \mathbb{R} \,\mid\, x \geq a \,\} \f$ and \f$\displaystyle (a, \infty) = \{\,x \in \mathbb{R} \,\mid\, x > a \,\} \) | ||
| $ \displaystyle \mathbb{C}$ | The set of complex numbers, $\displaystyle \mathbb{C} = \{\,a + b\,\mathrm{i} \mid a,\, b \in \mathbb{R}\,\}$ where $\mathrm{i}$ is the imaginary unit satisfying $\mathrm{i}^2 = -1$ | ||
| $ \displaystyle \mathbb{C}^*$ | The set of non-zero complex numbers, $\displaystyle \mathbb{C}^* = \mathbb{C} \setminus \{\,0\,\}$ | ||
| $ \displaystyle \mathbb{R}^n$ | The $n$-dimensional real coordinate space, where $n$ is a positive integer. This is basically the set containing all $n$-tuples of real numbers, defined by \( \mathbb{R}^n = \{\, (x_1, x_2, \ldots, x_n) \,\mid\, x_i \in \mathbb{R},\, i = \overline{1,n} \,\}. \) |