Countable and uncountable sets. There is a one-to- one correspondence between sets N (all natural numbers) and N 2 (even natural numbers). The correspondence has the form f ( n ) = 2 n . There is a one-to-one correspondence between sets N and N odd (odd natural numbers). The correspondence has the ...
Elements of mathematical proving. If p = ⇒ q ( ⋆ ) is a statement (let's call it a simple statement ), then there are 3 other statements related to this. q = ⇒ p - converse of statement ( ⋆ ) ; ( ∼ p ) = ⇒ ( ∼ q ) - inverse of statement ( ⋆ ) ; ( ∼ q ) = ⇒ ( ∼ p ) - contrapositive of statement ( ⋆ ...
Extremal elements Let _ be a partial order on P ⊂ X . Element a 0 ∈ P is the largest element in P , if a 0 is greater than every other element in P : ∀ a ∈ P ( a 6 = a 0 ) a ≺ a 0 . Element a 1 ∈ P is the smallest element in P , if a 1 is less than every other element in P : ∀ a ∈ P ( a 6 = a 1 ) a...
Function as a relation A subset R of the cartesian product X × Y is called a binary relation defined on elements of sets X and Y . Definition Let X and Y be any sets. A relation f which satisfies the following two conditions: ∀ x ∈ X ∃ y ∈ Y ( x, y ) ∈ f ; ∀ x ∈ X ∀ y 1 , y 2 ∈ Y [( x, y 1 ) ∈ f ∧ ...
General definitions of set Set is a basic concept that is not defined. Examples of sets: A = {1, 5, a,w,}; B = {x : x = 2 ∨ x = 5}; C = {∅, {1, 2}, 5}; D = {{{∅}}}. Sets: A,B,C. Elements of the set: a, b, c. The membership relation between set and its element: a ∈ A. If a is not element of A...
Generalization of union and intersection Definition: Let X 6 = ∅ be a space. Let R be a family of all subsets of the space X . Let T 6 = ∅ be an arbitrary set. Function f : T → R is called an indexed family of sets , if: ∀ t ∈ T ∃ A t ∈ R f ( t ) = A t . Examples: 1) Let T = { 1 , 2 , 3 , 4 , 5 } ,...
Graph of the relation Definition A directed graph is a set of points, together with a set of directed arcs connecting some of these points. The points of a directed graph are called vertices (singular: ver- tix) or the nodes of the graph. If D is a directed graph, then ν ( D ) is the set of vertice...
INCLUSION A set A is subset of a set B, if every element of A is also element of B. B is superset of A. A ⊂ B ⇔ ∀a(a ∈ A ⇒ a ∈ B). Examples: a) {1, 4, a} ⊂ {0, 1, 3, 4, a, b}; b) N ⊂ Z ⊂ Q ⊂ R - hierarchy of subsets. The empty set is a subset of any set: ∅ ⊂ A. A set is always a subset of it...
Logic connectives The logic connectives are: not ∼ and ∧ or ∨ if...then (implies) = ⇒ if and only if (equivalent) ⇐⇒ Examples of compound statements: ” Today is not Monday.” (truth-value = 1); 2 x = 2 if and only if x = 1 . (truth-value = 1); If x = 2 and y = 3 then x + y = 100 . (truth-value = 0) ...
Logic statement A statement is any sentence in logic sense that has a truth value , that is any sentence that is either true or false. The true - value is denoted by ”1”. The false - value is denoted by ”0”. Usually sentence are denoted by small letters, like p, q, r, s etc. Types of statements: si...
