Lectures in Set Theory - Lecture notes 1-5 PDF

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Lectures in Set Theory...

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Lectures in Set Theory Set Set is a collection of well – defined distinct objects. Every objects satisfied by the characteristics of a set is known as elements of a given set. The mathematical symbol “ ” (read as “a” is an element of set “A”) means that “a” belongs to set “A” or “a” is in “A”. While “ ” (read as “a” is not an element of set “A”) means that “a” is not belongs to set “A” or “a” is not in set “A”. Set was usually denoted by a capital letter (let say “A”, “B”, etc.) while the elements are inside a brace (e.g. {1, 2, e, r}). Sets maybe defined using Tabular/roster/listing down methods by enumerating every elements of a given set inside a brace while by Rule/set builder notation maybe done by giving a certain characteristics that can defined every elements of a set. Tabular method A = {1, 2, 3, 4, 5} B = {a, c, e, h, i, m, s, t} Rule method A = {x N | x < 6} B = {x | x is a letter of the word “MATHEMATICS”} Subset “

” (read as “B” is a subset of “A”) if every “x” elements of “B” (x

B) is

an elements of “A” (x A). “B” is an improper subset of “A” (in symbol ) if every “x” element of “A” is also in “B”. Ideally we can say that “B” is an improper subset of “A” if “A” is also a subset of “B”. Hence, if there exist “x” elements of “A” not found in “B” then “B” is a proper subset of “A”. The universal set (usually denoted by U) is a set whose elements are the collection of all the elements of all the given sets and may include elements which are not defined on those set. Mathematicians already enumerate the following pre-defined sets of numbers C = set of complex numbers (real plus imaginary numbers) (e.g. 2 + 3i) I = set of imaginary numbers (real multiply by

usually denoted or replaced

by i) (e.g. or 2i) R = set of Real numbers (rational or irrational) Q = set of rational numbers (numbers which maybe express as a ratio of two integers) (sets of repeating or terminating decimals) e.g. , Q’ = set of rational numbers (non repeating and non terminating decimals) e.g. e D = set of decimal numbers or fraction Z = set of integer (signed numbers including zero) + Z = set of positive integers Z = set of negative integers W = set of whole numbers (positive integers including zero) 1

N = set of natural numbers (positive integers) Cardinality of Set The cardinality of set “A” (in symbol |A| or n(A)) is the number of elements in “A”. Whereas using the our first example |A| or n(A) is equal to 5 (n(A) = 5). Type of set according to cardinality 1. Finite set ---- set those elements are countable of with limit or boundaries e.g. A = { 2, 3, 4, 7} B = {x | x...