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Description
Flashcards by Luke Danaher, updated more than 1 year ago
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Created by Luke Danaher
over 7 years ago
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| Question | Answer |
| Axiom | An axiom is a statement that mathematicians agree to treat as true. |
| Definition | A definition is a precise statement of the meaning of a mathematical word. |
| Even | A number n is even if and only if (iff) there exists an integer k such that n = 2k. |
| A Rational number | A rational number r, denoted by r ∈ Q, can be written as r = p/q, where p,q = 0 are integers, denoted by p,q ∈ Z |
| The Implication connective => | Given two statements P and Q. The assertion that if P is true then Q is true, denoted by P ⇒ Q or Q ⇐ P. We call P the Hypothesis and Q the Conclusion. |
| Converse | The statement ‘Q ⇒ P’ is the converse of the statement ‘P ⇒ Q’. |
| Logically equivalent | The statement ‘ ¬Q ⇒¬P’, is logically equivalent to ‘P ⇒ Q’ . |
| "iff" <=> | f ‘P ⇒ Q’ and ‘Q ⇒ P” then P ⇔ Q - read as ’P if and only if (iff) Q’ |
| Theorem | A theorem is a statement about mathematical objects. It consists of a hypothesis, or set of hypotheses, and a conclusion, or set of conclusions. |
| Proof | A proof of a theorem is a logical argument, a sequence of logical steps, that guarantees the conclusion holds when the hypothesis is satisfied |
| Proof by Contradiction | Proof by contradiction shows a statement A is true by first assuming that A is false. A logical argument with this hypothesis continues until a contradiction is arrived at. Thus A cannot be false and so must be true. |
| Proof by induction | Proof by Induction Proof by mathematical induction involves proving a statement P(n) is true for all ∀n ≥ n0, where n ∈ N the set of natural numbers (strictly positive integers). First we prove P(n0) is true, then assuming P(k) is true for n0 ≤ k ≤ n − 1, we deduce that P(n) is true. |
| A Set | A set is any well-defined collection of distinct elements. If x is an element of a set A, we write x ∈ A, otherwise we write x / ∈ A. |
| Cardinality | The number of elements of a set A is called its cardinality, denoted by |A| |
| Integers | Z = {...,−2,−1,0,1,2,...} |
| Natural numbers | N = Z+ = {1,2,3,...} |
| Rational numbers | The set of rational numbers, Q, is defined by Q = {p/q | p,q ∈ Z, q = 0}. |
| Equal sets | Two sets A and B are said to be equal, denoted by A = B, if they contain precisely the same elements, i.e. A = B means x ∈ A ⇐⇒ x ∈ B. |
| The Universal set | The universal set, denoted by U, is the set that contains all possible objects of interest. |
| The empty set | The empty set, denoted by ∅, is the unique set with no elements and zero cardinality. |
| A subset | A set A is a subset of a set B, denoted by A ⊆ B or B ⊇ A, if every element of a set A is also an element of a set B. |
| A proper subset | A set A is a proper subset of B, denoted by A ⊂ B, if A ⊆ B and A does not equal B, A = ∅. |
| Set Intersection | Given two sets A and B, the intersection of A and B, denoted by A ∩ B, is the set of all elements which are elements of both A and B. Thus A ∩ B = B ∩ A = {x ∈ U | x ∈ A and x ∈ B} |
| Distinct/disjoint sets | We say two sets are distinct, or disjoint, if A ∩ B = ∅ (no elements in common). |
| Set union | Given two sets A and B, the union of A and B, denoted by A ∪ B, is the set of all elements that are elements of either A or B, or both. Thus A ∪ B = B ∪ A = {x ∈ U | x ∈ A or x ∈ B} |
| Set difference | Given two sets A and B, the difference of A and B, denoted by A \ B, is the set of all elements that are elements of A but not of B. Thus A \ B = {x ∈ U | x ∈ A and x / ∈ B} |
| Relative complement of sets | The difference A \ B, is called the (relative) complement of B in A. Given a universal set U |
| The absolute complement of sets | The (absolute) complement of a set A ⊂ U, denoted by Ac, is the set of all elements of U that are not elements of A. Ac = U \ A = {x ∈ U | x / ∈ A} =⇒ (Ac) c = A |
| Cartesian Product | The Cartesian product of two sets X and Y is the set, denoted by X × Y , whose elements are ordered pairs of the form X × Y = {(x,y) |x ∈ X and y ∈ Y } |
| The real numbers | R denotes the set of real numbers consisting of both rational and irrationals. |
| A Field (F) | A field F is a set of numbers with two operations addition (+) and multiplication (×). {see axioms in n, s&s). |
| Ordered fields | A set of numbers F is said to be ordered if they satisfy the ordering axioms (see n,s&s) |
| Upper/lower bounds of sets | We say B is an upper (lower) bound of a set X if ∀x ∈ X, x ≤ B (x ≥ B). A set X is bounded if it has both a lower and upper bound. |
| Supremum of a set | The least upper bound is known as the supremum |
| A complete set | An ordered field F is said to complete if every non-empty bounded subset has a supremum in F |
| Smallest ordered and complete field | R is the smallest set of numbers which is a complete, ordered field. |
| Interval | |
| Modulus function | |
| Triangle Inequality | x,y ∈ F ||x| − |y|| ≤ |x ± y| ≤ |x| + |y| |
| e-neighbourhood | Given ε > 0, an ε-neighbourhood Nε(c) of a point c ∈ R is the set of points Nε(c) := (c − ε,c + ε) = {x ∈ R | |x − c| < ε} ⊂ R |
| Deleted/punctured e-neighbourhood | Given ε > 0, a deleted, or punctured, ε-neighbourhood Nε(c)\ {c} of a point c ∈ R is the set of points Nε(c) \ {c} := (c − ε,c) ∪ (c,c + ε) = {x ∈ R | 0 < |x − c| < ε} ⊂ R |
| Interior point | A point c ∈ R is called an interior point of a set S ⊆ R if ∃ ε > 0 such that the set of points Nε(c) ⊂ S. The set of all interior points of S is called the interior of S and is denoted by S0. |
| Exterior point | A point c ∈ R is called an exterior point of a set S ⊆ R if ∃ ε > 0 such that the set of points Nε(c) ⊂ Sc = R \ S. |
| Boundary point | A point c ∈ R is called an boundary point of a set S ⊂ R if every ε-neighbourhood of c contains both points of S and Sc, or ∀ ε > 0, Nε(c) ∩ S = ∅ and Nε(c) ∩ Sc = ∅. The set of all boundary points of S is called the boundary of S and is denoted by ∂S. |
| Limit point | A point c ∈ R is called an limit point of a set S ⊆ R if every deleted ε-neighbourhood of c contains a point a ∈ S, or ∀ ε > 0, ∃ a ∈ S such that 0 < |c − a| < ε, or Nε(c) \ {c} ∩ S = ∅. The set of all limit points of S is denoted by S′. |
| Closure point & closure of a set | A point c ∈ R is called a closure point of a set S ⊆ R if every ε-neighbourhood of c contains a point of S, or ∀ ε > 0, Nε(c) ∩ S = ∅. The set of all closure points of S is called the closure of S and is denoted by ¯ S |
| Topology identities | S0 ⊆ S,S′ ⊆ ¯ S or ¯ S = S0 ∪ ∂S = S ∪ S′, ∂S = ¯ S \ S0 |
| Open set | A set S ⊆ R is said to be an open set if it contains only interior points, S = S0, or ∀ x ∈ S, ∃ ε > 0 such that Nε(x) ⊂ S. |
| Closed set | A set S ⊆ R is said to be closed if it contains all its limit points, S′ ⊆ S, or if it has no limit points, S′ = ∅, so in both cases S = ¯ S. |
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