Number Sets

Natural numbers
|(\mathbb N)|

Numbers used for counting
|\mathbb{N} = \{0,1,2,3,4,5,\ldots\}|

|3,\ ||5,\ ||134,\ ||2099|

Integers
|(\mathbb Z)|

Natural numbers and their opposites
|\mathbb{Z} = \{\ldots,-3,-2,-1,0,1,2,3,\ldots\}|

|-133,\ ||-9,\ ||0,\ ||9,\ ||915|

Rational numbers
|(\mathbb Q)|

Numbers that can be expressed as the ratio of 2 integers

|-5.\overline{3},\ ||-\dfrac{1}{3},\ ||\dfrac{3}{4},\ ||3,\ ||6.4|

Irrational numbers
|(\mathbb{R}\backslash\mathbb{Q}| or |\mathbb{Q}')|

Numbers that cannot be expressed as the ratio of 2 integers

|\sqrt{2},\ ||\sqrt[3]{11},\ ||e,\ ||\pi|

Real numbers
|(\mathbb R)|

Numbers that belong to either the set of rational numbers or the set of irrational numbers

|-16,\ ||\dfrac{5}{8},\ ||\sqrt{5},\ ||7,\ ||23.\overline{6}|

​|\{\ ,\,\}|

Listing the elements of a set in expanded form

|\{0,1,2,3,4,...\}|

​|\in|

"... belongs to..." or "... is an element of..."

|3\in \mathbb{N}|

​|\notin|

"... does not belong to..." or  "... is not an element of..."

|\mathbb{Z}\subset\mathbb{Q}|

|\subset|

"... is included in..." or  "... is a subset of..."

|\mathbb{Z}\subset\mathbb{Q}|

|\large\cup|

Union of 2 sets

|\mathbb{R}=\mathbb{Q} \cup \mathbb{Q’}|

|\tiny\boxed{\phantom{\normalsize{t}}}^{\large*}|

Excluding the number zero

|\mathbb{N}^*| represents the set of natural numbers excluding |0.|

|\tiny\boxed{\phantom{\normalsize{t}}}_{\large+}|

Only the positive elements of the set

|\mathbb{Z}_+| represents the set of positive integers.

|\tiny\boxed{\phantom{\normalsize{t}}}_{\large-}|

Only the negative elements of the set

|\mathbb{Q}_-| represents the set of only negative rational numbers.