Cheat Sheet of Mathemtical Notation and Terminology
Logic and Sets
Notation Terminology Explanation and Examples
𝑎∶= 𝑏defined by The object 𝑎on the side of the colon is defined by 𝑏.
Examples:𝑥∶= 5 means that 𝑥is defined to be 5, or 𝑓(𝑥) ∶= 𝑥2− 1 means that
the function 𝑓is defined to be 𝑥2− 1, or 𝐴∶= {1,5,7} means that the set 𝐴is
defined to be {1,5.7}.
𝑆1⇒𝑆2implies Logical implication: If statement 𝑆1is true, then statement 𝑆2must be tr ue. We
say 𝑆1is a sufficient condition for 𝑆2or 𝑆2is a necessary condition for 𝑆1.
Examples:(𝑛∈ℕeven) ⇒(𝑛2even).
𝑆1⇔𝑆2equivalent to Logical equivalence: If statement 𝑆1is true, then statement 𝑆2must be true, and
vice versa. We say 𝑆2is a necessary and sufficient condition for 𝑆1.
Examples:(ln 𝑥 > 0) ⇔(𝑥 > 1).
∃there exists Abbreviation for there exists
∀for all Abbreviation for for all
{… } set The “objects” listed between the curly brackets are members of the set being
defined.
Examples:{0,2,5,7},{2 + 𝑖, 7 − √5},{,,}
The elements of a set can be any kind of objects such as numbers, functions,
points, geometric objects or other.
𝑎∈𝐴element of 𝑎is an element of the set 𝐴, that is, 𝑎is in the set 𝐴.
Examples:𝜋∈ℝ,4 ∈ {1,4,7},∈{, , }
∅or {} empty set The special set that does not contain any element.
{𝑥∣property}set of .. . with .. . Notation indicating a set of elements 𝑥satisfying a certain property.
Examples:{𝑛∈ℕ∣𝑛is even}, where 𝑛∈ℕis the typical element and the
property satisfied is that 𝑛is even.
{𝑥2∣𝑥∈ℕ}, where the typical member is a square of some number in ℕ.
𝐴 ⊆ 𝐵 subset of The set 𝐴is a subset of 𝐵, that is, every element of 𝐴is also an element of 𝐵.
More formally: 𝑏∈𝐵⇒𝑏∈𝐴.
Examples:ℚ⊆ℝ,{1,4,7} ⊆{1,2,3,4,5,6,7}
𝐴∪𝐵union The set of elements either in 𝐴or in 𝐵.
More formally: (𝑥∈𝐴∪𝐵)⇔(𝑥∈𝐴or 𝑥∈𝐵).
Examples:{1,4,7} ∪ {4,5,8} = {1,4,5,7,8} (elements are not repeated in a
union if they appear in both sets!)
Note:We can look at a union of an arbitrary collection of sets: The set of objects
that appear in at least one of the sets in the collection.
𝐴∩𝐵intersection The set of elements that are in 𝐴and in 𝐵.
More formally: (𝑥∈𝐴∩𝐵)⇔(𝑥∈𝐴and 𝑥∈𝐵).
Examples:{1,4,7} ∩ {1,2,3,5,6,7} = {1,7}
Note:We can look at the intersection of an arbitrary collection of sets: The set of
objects that appear in every set in the collection.
𝐴⧵𝐵complement The set of elements that are in 𝐴but not in 𝐵.
More formally: (𝑥∈𝐴⧵𝐵)⇔(𝑥∈𝐴and 𝑥∉𝐵).
Examples:{1,4,5,7} ⧵{1,2,3,6,7} = {4,5}
Interval notation
Notation Terminology Explanation and Examples
[𝑎, 𝑏]closed interval If 𝑎, 𝑏 ∈ℝwith 𝑎≤𝑏the closed inter valis the set {𝑥∈ℝ∣𝑎≤𝑥≤𝑏}
Examples:[−3,5] is the set of real numbers between −3 and 5, including the
endpoints −3 and 5.
(𝑎, 𝑏)open interval If 𝑎, 𝑏 ∈ℝwith 𝑎≤𝑏the open inter valis the set {𝑥∈ℝ∣𝑎<𝑥<𝑏}
Examples:(−3,5) id the set of real numbers between −3 and 5, excluding the
endpoints −3 and 5.
[𝑎, 𝑏)or (𝑎, 𝑏]half open interval If 𝑎, 𝑏 ∈ℝwith 𝑎≤𝑏,[𝑎,𝑏)is the set of all numbers between 𝑎and 𝑏with
𝑎included and 𝑏excluded. In case of (𝑎, 𝑏]the endpoint 𝑎is excluded and 𝑏
is included.
Examples:[−3,5) is the set of real numbers between −3 and 5, including −3
but excluding 5. For (−3,5] the endpoint −3 is excluded and 5is included.
[𝑎, ∞) or (−∞, 𝑎]closed half line If 𝑎∈ℝ, then [𝑎, ∞) is the set of real numbers larger than or equal to 𝑎, and
(−∞, 𝑎]is the set of real numbers less than or equal to 𝑎
(𝑎, ∞) or (−∞, 𝑎)open half line If 𝑎∈ℝ, then (𝑎, ∞) is the set of real numbers strictly larger than 𝑎, and
(−∞, 𝑎)is the set of real numbers strictly less than 𝑎
Examples:(0,∞) set of all positive real numbers; (−∞,5] set of all real num-
bers less than or equal to 5.
Functions
Notation Terminology Explanation and Examples
𝑓∶𝐴→𝐵function A function 𝑓from the set 𝐴to the set 𝐵is a rule that assigns every element
𝑥∈𝐴a unique element 𝑓(𝑥) ∈ 𝐵.
The set 𝐴is called the domain and represents all possible (or desirable) “in-
puts”, the set 𝐵is called the codomain and contains all potential “outputs”.
𝑥→ 𝑓(𝑥)is mapped to The function maps 𝑥to the value 𝑓(𝑥).
Examples:𝑔∶ℝ→ℂ,𝜃→ 𝑔(𝜃) ∶= 𝑒𝑖𝜃. A function from ℝto ℂgiven by 𝑒𝑖𝜃;
𝑓∶ℝ→ℝ,𝑥→ 𝑓(𝑥) ∶= 1 + 𝑥2. A function from ℝto ℝgiven by 1 + 𝑥2;
ℎ∶ℂ→[0,∞),𝑧→ ℎ(𝑧) ∶= |𝑧|. A function from ℂto [0,∞) given by |𝑧|.
im(𝑓)image or range The set of values 𝑓∶𝐴→𝐵attains: im(𝑓) ∶= {𝑓(𝑥)∶ 𝑥∈𝐴}⊆ 𝐵.
Examples:𝑓∶ℝ→ℝ,𝑥→ 𝑓(𝑥) ∶= 𝑥2. The codomain is ℝ, the image or
range is [0,∞).
surjective or onto A function 𝑓∶𝐴→𝐵is called surjective if im(𝑓) = 𝐵, that is, the codomain
coincides with the range.
More formally: For every 𝑏∈𝐵there exists 𝑎∈𝐴such that 𝑓(𝑎) = 𝑏.
Note:𝑓∶𝐴→im(𝑓)is always surjective. The choice of codomain is quite
arbitrary. We often just state the general objects rather than the image or range.
For instance function values are in ℝif we are not intested in the image.
injective or one-to-one A function 𝑓∶𝐴→𝐵is called injective if im(𝑓) = 𝐵, that is, every point in
the image comes from exactly one point in the domain 𝐴.
More formally: If 𝑎1, 𝑎2∈𝐴are such that 𝑓(𝑎1) = 𝑓(𝑎2), then 𝑎1=𝑎2.
bijective A function 𝑓∶𝐴→𝐵is called bijective if it is surjective and injective.
𝑓−1 inverse function A function 𝑓∶𝐴→𝐵is called invertible if it is bijective. The inverse func-
tion 𝑓−1 ∶𝐵→𝐴is defined as follows: Given 𝑏∈𝐵take the unique point
𝑎∈𝐴such that 𝑓(𝑎) = 𝑏and set 𝑓−1(𝑏) ∶= 𝑎(by surjectivity such 𝑎exists,
by injectivity it is unique).