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MathsNotations

set theory

Set theory notations

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} \,\}.

Last Modified: 27 Dec 11 @ 02:14     Page Rendered: 2022-03-14 15:41:44