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Cheat Sheet of Mathemtical Notation and Terminology, Cheat Sheet of Mathematics

In this useful cheat sheet you have Notation, Terminology, Explanation and Examples on Logic and Sets, Interval notation and Functions.

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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}.
𝑆1𝑆2implies 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:(𝑛even) (𝑛2even).
𝑆1𝑆2equivalent 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.
𝑎𝐴element of 𝑎is an element of the set 𝐴, that is, 𝑎is in the set 𝐴.
Examples:𝜋,4 {1,4,7},{, , }
or {} empty set The special set that does not contain any element.
{𝑥property}set of .. . with .. . Notation indicating a set of elements 𝑥satisfying a certain property.
Examples:{𝑛𝑛is even}, where 𝑛is the typical element and the
property satisfied is that 𝑛is even.
{𝑥2𝑥}, where the typical member is a square of some number in .
𝐴 𝐵 subset of The set 𝐴is a subset of 𝐵, that is, every element of 𝐴is also an element of 𝐵.
More formally: 𝑏𝐵𝑏𝐴.
Examples:,{1,4,7} {1,2,3,4,5,6,7}
𝐴𝐵union The set of elements either in 𝐴or in 𝐵.
More formally: (𝑥𝐴𝐵)(𝑥𝐴or 𝑥𝐵).
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.
𝐴𝐵intersection The set of elements that are in 𝐴and in 𝐵.
More formally: (𝑥𝐴𝐵)(𝑥𝐴and 𝑥𝐵).
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.
𝐴𝐵complement The set of elements that are in 𝐴but not in 𝐵.
More formally: (𝑥𝐴𝐵)(𝑥𝐴and 𝑥𝐵).
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 with 𝑎𝑏the closed inter valis the set {𝑥𝑎𝑥𝑏}
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 with 𝑎𝑏the open inter valis the set {𝑥𝑎<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 with 𝑎𝑏,[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, 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, 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
𝑓𝐴𝐵function A function 𝑓from the set 𝐴to the set 𝐵is a rule that assigns every element
𝑥𝐴a 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,𝜃→ 𝑔(𝜃) ∶= 𝑒𝑖𝜃. A function from to given by 𝑒𝑖𝜃;
𝑓,𝑥→ 𝑓(𝑥) ∶= 1 + 𝑥2. A function from to given by 1 + 𝑥2;
[0,∞),𝑧→ (𝑧) ∶= |𝑧|. A function from to [0,∞) given by |𝑧|.
im(𝑓)image or range The set of values 𝑓𝐴𝐵attains: im(𝑓) ∶= {𝑓(𝑥)∶ 𝑥𝐴} 𝐵.
Examples:𝑓,𝑥→ 𝑓(𝑥) ∶= 𝑥2. The codomain is , the image or
range is [0,∞).
surjective or onto A function 𝑓𝐴𝐵is called surjective if im(𝑓) = 𝐵, that is, the codomain
coincides with the range.
More formally: For every 𝑏𝐵there exists 𝑎𝐴such that 𝑓(𝑎) = 𝑏.
Note:𝑓𝐴im(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 if we are not intested in the image.
injective or one-to-one A function 𝑓𝐴𝐵is 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, 𝑎2𝐴are such that 𝑓(𝑎1) = 𝑓(𝑎2), then 𝑎1=𝑎2.
bijective A function 𝑓𝐴𝐵is called bijective if it is surjective and injective.
𝑓−1 inverse function A function 𝑓𝐴𝐵is called invertible if it is bijective. The inverse func-
tion 𝑓−1 𝐵𝐴is defined as follows: Given 𝑏𝐵take the unique point
𝑎𝐴such that 𝑓(𝑎) = 𝑏and set 𝑓−1(𝑏) ∶= 𝑎(by surjectivity such 𝑎exists,
by injectivity it is unique).

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Cheat Sheet of Mathemtical Notation and Terminology

Logic and Sets

Notation Terminology Explanation and Examples 푎 ∶= 푏 defined by The object 푎 on the side of the colon is defined by 푏. Examples: 푥 ∶= 5 means that 푥 is defined to be 5 , or 푓 (푥) ∶= 푥^2 − 1 means that the function 푓 is defined to be 푥^2 − 1, or 퐴 ∶= {1, 5 , 7} means that the set 퐴 is defined to be {1, 5 .7}. 푆 1 ⇒ 푆 2 implies Logical implication: If statement 푆 1 is true, then statement 푆 2 must be true. We say 푆 1 is a sufficient condition for 푆 2 or 푆 2 is a necessary condition for 푆 1. Examples: (푛 ∈ ℕ even) ⇒ (푛^2 even). 푆 1 ⇔ 푆 2 equivalent to Logical equivalence: If statement 푆 1 is true, then statement 푆 2 must be true, and vice versa. We say 푆 2 is a necessary and sufficient condition for 푆 1. Examples: (ln 푥 > 0) ⇔ (푥 > 1). ∃ there exists Abbreviation for there existsfor 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 + 푖, 7 −

The elements of a set can be any kind of objects such as numbers, functions, points, geometric objects or other. 푎 ∈ 퐴 element of 푎 is an element of the set 퐴, that is, 푎 is in the set 퐴.

Examples: 휋 ∈ ℝ, 4 ∈ {1, 4 , 7}, ∈ , , Á or {} empty set The special set that does not contain any element. {푥 ∣ property} set of... with... Notation indicating a set of elements 푥 satisfying a certain property. Examples: {푛 ∈ ℕ ∣ 푛 is even}, where 푛 ∈ ℕ is the typical element and the property satisfied is that 푛 is even. {푥^2 ∣ 푥 ∈ ℕ}, where the typical member is a square of some number in ℕ. 퐴 ⊆ 퐵 subset of The set 퐴 is a subset of 퐵, that is, every element of 퐴 is also an element of 퐵. More formally: 푏 ∈ 퐵 ⇒ 푏 ∈ 퐴. Examples: ℚ ⊆ ℝ, {1, 4 , 7} ⊆ {1, 2 , 3 , 4 , 5 , 6 , 7} 퐴 ∪ 퐵 union The set of elements either in 퐴 or in 퐵. More formally: (푥 ∈ 퐴 ∪ 퐵) ⇔ (푥 ∈ 퐴 or 푥 ∈ 퐵). 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. 퐴 ∩ 퐵 intersection The set of elements that are in 퐴 and in 퐵. More formally: (푥 ∈ 퐴 ∩ 퐵) ⇔ (푥 ∈ 퐴 and 푥 ∈ 퐵). 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. 퐴 ⧵ 퐵 complement The set of elements that are in 퐴 but not in 퐵. More formally: (푥 ∈ 퐴 ⧵ 퐵) ⇔ (푥 ∈ 퐴 and 푥 ∉ 퐵). Examples: {1, 4 , 5 , 7} ⧵ {1, 2 , 3 , 6 , 7} = {4, 5}

Interval notation

Notation Terminology Explanation and Examples [푎, 푏] closed interval If 푎, 푏 ∈ ℝ with 푎 ≤ 푏 the closed interval is the set {푥 ∈ ℝ ∣ 푎 ≤ 푥 ≤ 푏} Examples: [−3, 5] is the set of real numbers between −3 and 5 , including the endpoints −3 and 5. (푎, 푏) open interval If 푎, 푏 ∈ ℝ with 푎 ≤ 푏 the open interval is the set {푥 ∈ ℝ ∣ 푎 < 푥 < 푏} Examples: (−3, 5) id the set of real numbers between −3 and 5 , excluding the endpoints −3 and 5. [푎, 푏) or (푎, 푏] half open interval If 푎, 푏 ∈ ℝ with 푎 ≤ 푏, [푎, 푏) is the set of all numbers between 푎 and 푏 with 푎 included and 푏 excluded. In case of (푎, 푏] the endpoint 푎 is excluded and 푏 is included. Examples: [−3, 5) is the set of real numbers between −3 and 5 , including − but excluding 5. For (−3, 5] the endpoint −3 is excluded and 5 is included. [푎, ∞) or (−∞, 푎] closed half line If 푎 ∈ ℝ, then [푎, ∞) is the set of real numbers larger than or equal to 푎, and (−∞, 푎] is the set of real numbers less than or equal to 푎 (푎, ∞) or (−∞, 푎) open half line If 푎 ∈ ℝ, then (푎, ∞) is the set of real numbers strictly larger than 푎, and (−∞, 푎) is the set of real numbers strictly less than 푎 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 푓 ∶ 퐴 → 퐵 function A function 푓 from the set 퐴 to the set 퐵 is a rule that assigns every element 푥 ∈ 퐴 a unique element 푓 (푥) ∈ 퐵. The set 퐴 is called the domain and represents all possible (or desirable) “in- puts”, the set 퐵 is called the codomain and contains all potential “outputs”. 푥 → 푓 (푥) is mapped to The function maps 푥 to the value 푓 (푥). Examples: 푔 ∶ ℝ → ℂ, 휃 → 푔(휃) ∶= 푒푖휃^. A function from ℝ to ℂ given by 푒푖휃^ ; 푓 ∶ ℝ → ℝ, 푥 → 푓 (푥) ∶= 1 + 푥^2. A function from ℝ to ℝ given by 1 + 푥^2 ; ℎ ∶ ℂ → [0, ∞), 푧 → ℎ(푧) ∶= 푧. A function from ℂ to [0, ∞) given by 푧. im(푓 ) image or range The set of values 푓 ∶ 퐴 → 퐵 attains: im(푓 ) ∶= {푓 (푥) ∶ 푥 ∈ 퐴} ⊆ 퐵. Examples: 푓 ∶ ℝ → ℝ, 푥 → 푓 (푥) ∶= 푥^2. The codomain is ℝ, the image or range is [0, ∞). surjective or onto A function 푓 ∶ 퐴 → 퐵 is called surjective if im(푓 ) = 퐵, that is, the codomain coincides with the range. More formally: For every 푏 ∈ 퐵 there exists 푎 ∈ 퐴 such that 푓 (푎) = 푏. Note: 푓 ∶ 퐴 → im(푓 ) 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 ℝ if we are not intested in the image. injective or one-to-one A function 푓 ∶ 퐴 → 퐵 is called injective if im(푓 ) = 퐵, that is, every point in the image comes from exactly one point in the domain 퐴. More formally: If 푎 1 , 푎 2 ∈ 퐴 are such that 푓 (푎 1 ) = 푓 (푎 2 ), then 푎 1 = 푎 2. bijective A function 푓 ∶ 퐴 → 퐵 is called bijective if it is surjective and injective. 푓 −1^ inverse function A function 푓 ∶ 퐴 → 퐵 is called invertible if it is bijective. The inverse func- tion 푓 −1^ ∶ 퐵 → 퐴 is defined as follows: Given 푏 ∈ 퐵 take the unique point 푎 ∈ 퐴 such that 푓 (푎) = 푏 and set 푓 −1(푏) ∶= 푎 (by surjectivity such 푎 exists, by injectivity it is unique).