Cardinality of Reals
Consider the cardinality of the real numbers. Here we will prove the real numbers are an infinite set that are uncountable. Let $f(x)=\frac{1}{1+n}| n \in \mathbb{N}.$ Here you can see that all the fractions in this sequence are between 0 and 1. It is also apparent that each fraction corresponds to a natural number. In that sense each natural number has been paired with a fraction. There are as many fractions as there are natural numbers. Yet this set of fractions is only a subset of the real numbers, therefore, the real numbers are not countably infinite. The real numbers...
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This page discusses cardinality, a basic topic found in classes like Sets, Relations and Functions and also Real Analysis. Definitions with discussion follow. numerically equivalent two sets are numerically equivalent if there exists a one-to-one function from one set onto the other. We can also say two sets have the same cardinal number if this is true. </dl> If we can say that a set has $n$ elements where $n$ is a specific element of the natural numbers, then we can say that particular set is countable. If each element in a infinite set can be paired with a natural...
read moreCauchy Sequences
A sequence $a_n$ converges to $a$ if for every $\epsilon > 0$, $\exists$ an $N$ such that if $n > N$ then $\left| a_n - a \right| < \epsilon.$
read moreLogic and Quantifiers
This page discusses Logic and Quantifiers found in many Real Analysis and logic classes. Definitions with discussion follow. proposition an assertion that is either true or false but not both. A proposition is called simple or atomic if it is made up of only one proposition. A proposition is compound if it consists of one or more simple propositions with one or more logical connectives. logical connective joins simple propositions together to create a new proposition which may differ in truth value truth value the validity of a proposition. The truth value of a statement may be true or false...
read moreNumber Systems
This page discusses the types of number systems found in real analysis and introduction to proving classes. Number System is a set of numbers together with one or more operations. Symbol Name Set $\mathbb{N}$ Natural Numbers {1, 2, 3, ...} $\mathbb{Z}$ Integers {...,-3, -2, -1, 0, 1, 2, 3, ...} $\mathbb{Q}$ Rational Numbers $\{\frac{a}{b}| a,b \in \mathbb{Z}\}$. All fractions. $\mathbb{R} - \mathbb{Q}$ Irrational Numbers Numbers which cannot be expressed in the form of a fraction. $\mathbb{R}$ Real Numbers Satisfies the Real Number Axioms. All finite and infinite decimal numbers. $\mathbb{C}$ Complex Numbers $\{a+bi \ \big| \ a,b \in \mathbb{R} \text{...
read moreNumber Theory
Factor to factor means to write as a product Factor 6 $ 6 = 2 \cdot 3 $ When factoring, it is often useful to list only the prime factors. Usually, we do not list 1 as a factor. Prime A prime number is a natural number (positive, whole) only divisible by 1 and itself. A prime number must also be greater than 1. 2 is the first prime number. 2 is the only even prime number. Composite A composite number is a natural number greater than 1 that is not prime. Factor 26 $ 2 \cdot 13 $ Factor...
read moreOrdered Fields
A ring is a set $R$ with the usual operations of multiplication and addition from $R \times R \rightarrow R$ which satisfy the properties of: associativity under addition commutativity under addition the existence of additive identity the existence of an additive inverse associativity under multiplication distributivity across addition These properties are defined in the above mentioned link to the page real number axioms.
read moreSets, Relations and Funcs
This page discusses Sets,Relations and Function. The ideas presented here only skim the surface. Definitions and discussion follow. relation A subset $R$ of $A \times B$ is a relation from $A$ to $B$. inverse If $R$ is $\{(a, b) | (a,b) \in R\}$, then $R^{-1} = \{(b, a) | (a, b) \in R\}$ function a mapping of $f$ from $X$ into $Y$ such that if $(x,y_1) \in f$ and $(x,y_2) \in f$, then $y_1=y_2$. To designate this we write $f:X \rightarrow Y$. onto if f is a function from $X$ to $Y,$ and the range(see below) of $f$ is $Y$...
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This page discusses sets, a topic that forms the foundation of arithmetic and real analysis alike. set a collection element a member of a set. We denote $x$ is a element of set $S$ by $x \in S$ empty set, $\varnothing$ the set with no elements subset a set $S$ is a subset of $T$, written $S \subset T$, if $ \forall x$ if $ x \in S \Rightarrow x \in T $ finite set a set in which each element can be matched with exactly one element of the set $\{1, 2, ..., n\}$ set equality two sets $S$...
read moreThe Real Number Axioms
The set of real numbers, $\mathbb{R}$, satisfies the following three sets of axioms.
read moreThe Real Number System
The real number system, $\mathbb{R}$, is a ring, $R,$ definined on the quotient set of all rational Cauchy sequences such that the operations of addition and multiplication are defined as follows: $$[a_n] + [b_n] = [a_n + b_n] \\ [a_n][b_n] = [a_n b_n] $$
read moreTypes of Numbers
Common types of numbers found in Arithmetic. Integer any whole number including 0 such as 1, 2, or 3. $\frac{1}{2}$ is not a integer. $0.333\bar{3}$ is not an integer. $\frac{6}{3}$ is an integer even though it is a fraction because it reduces to 2 which is an integer. Negative whole numbers are also integers. The symbol for integers in mathematics is $\mathbb{Z}$ named after the German word 'zahlen.' Even Number Any number, $n$, that can be expressed as a multiple of 2. Now, 0 is an even number because $0 = 2 \cdot 0$. Negative numbers which are multiples of...
read moreUnary and Binary Ops
This page discusses unary and binary operations found in many Sets, Relations and Functions classes and also advanced Real Analysis. unary operation a mathematical function, say u, mapping a set S on to itself, i.e. $u:S \rightarrow S.$ binary operation a procedure carried out on two numbers. In particular, it is a function that maps the cartesian product of two elements in a set on to itself, i.e. $S \times S \rightarrow S.$ algebraic structure is a set, say S, along with a finite number of operations. Example: $ \{ \mathbb{Z^{+}},- \}.$ closed or closed under Let * be a...
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