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

Download Cheat Sheet of Mathemtical Notation and Terminology and more Cheat Sheet Mathematics in PDF only on Docsity! Cheat Sheet of Mathemtical Notation and Terminology Logic and Sets Notation Terminology Explanation and Examples a ∶= b defined by The object a on the side of the colon is defined by b. Examples:x ∶= 5 means that x is defined to be 5, or f (x) ∶= x2 − 1 means that the function f is defined to be x2 − 1, or A ∶= {1, 5, 7} means that the set A is defined to be {1, 5.7}. S1 ⇒ S2 implies Logical implication: If statement S1 is true, then statement S2 must be true. We say S1 is a sufficient condition for S2 or S2 is a necessary condition for S1. Examples:(n ∈ ℕ even) ⇒ (n2 even). S1 ⇔ S2 equivalent to Logical equivalence: If statement S1 is true, then statement S2 must be true, and vice versa. We say S2 is 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 ∈ A element of a is an element of the set A, that is, a is in the set A. Examples: ∈ 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 x satisfying a certain property. Examples:{n ∈ ℕ ∣ n is even}, where n ∈ ℕ is the typical element and the property satisfied is that n is even. {x2 ∣ x ∈ ℕ}, where the typical member is a square of some number in ℕ. A ⊆ B subset of The set A is a subset of B, that is, every element of A is 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 ∪ B union The set of elements either in A or in B. More formally: (x ∈ A ∪ B) ⇔ (x ∈ A or 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 ∩ B intersection The set of elements that are in A and in B. More formally: (x ∈ A ∩ B) ⇔ (x ∈ A and 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 ⧵ B complement The set of elements that are in A but not in B. More formally: (x ∈ A ⧵ B) ⇔ (x ∈ A and 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 ∈ ℝ with a ≤ b the closed interval is 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 ∈ ℝ with a ≤ b the open interval is 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 ∈ ℝ with a ≤ b, [a, b) is the set of all numbers between a and b with a included and b excluded. In case of (a, b] the endpoint a is 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 5 is 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 → B function A function f from the set A to the set B is a rule that assigns every element x ∈ A a unique element f (x) ∈ B. The set A is called the domain and represents all possible (or desirable) “in- puts”, the set B is called the codomain and contains all potential “outputs”. x → f (x) is mapped to The function maps x to the value f (x). Examples:g∶ ℝ → ℂ,  → g() ∶= ei. A function from ℝ to ℂ given by ei; f ∶ ℝ → ℝ, x → f (x) ∶= 1 + x2. A function from ℝ to ℝ given by 1 + x2; ℎ∶ ℂ → [0,∞), z → ℎ(z) ∶= |z|. A function from ℂ to [0,∞) given by |z|. im(f ) image or range The set of values f ∶ A → B attains: 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 → B is called surjective if im(f ) = B, that is, the codomain coincides with the range. More formally: For every b ∈ B there exists a ∈ A such 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 ℝ if we are not intested in the image. injective or one-to-one A function f ∶ A → B is 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 ∈ A are such that f (a1) = f (a2), then a1 = a2. bijective A function f ∶ A → B is called bijective if it is surjective and injective. f−1 inverse function A function f ∶ A → B is called invertible if it is bijective. The inverse func- tion f−1∶ B → A is defined as follows: Given b ∈ B take the unique point a ∈ A such that f (a) = b and set f−1(b) ∶= a (by surjectivity such a exists, by injectivity it is unique).