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