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# set theory

Set theory notations

## Set Theory

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

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

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