Math Squad infinity review of set theory function mhossain S13 - Rhea

Math Squad

↳ To Infinity and Beyond. Introduction

↳ Review of Set Theory

↳ Functions

↳ Countability

↳ Cardinality

↳ Hilbert's Grand Hotel

Let $ f $ be a function from set $ A $ to set $ B $.

$ \Leftrightarrow f : A \rightarrow B $

Onto (Surjective)

Definition: The function $ f $ is said to be surjective (or to map $ A $ onto $ B $) if $ f(A) = B $; that is, if the range $ F(f) = B $. If $ f $ is a surjective function, we also say that $ f $ is a surjection.

In other words, if for every $ y $ in $ B $, there exists at least one $ x $ in $ A $ such that $ f(x) = y $, then $ f $ us onto $ B $.

e.g. Given that

$ f(x) = x^5, \forall x \in \mathbb{R}, f : \mathbb{R} \rightarrow \mathbb{R} $

then, $ f $ is onto the set R.

e.g. Given that

$ f(x) = x^2, \forall x \in \mathbb{R}, f : \mathbb{R} \rightarrow \mathbb{R} $

then, $ f $ is not onto the set R.

But if we define the domain of $ f $ such that

$ f(x) = x^2, \forall x \in \mathbb{R}, f : \mathbb{R} \rightarrow [0,\infty] $

then, $ f $ is not onto.

Figures one and two show some other examples of functions, one of which is onto and one which is not.

$ y = f(x) = x+5x^2+x^3, f:\mathbb{R}\rightarrow\mathbb{R} $ Fig 1: $ f $ is onto since every value on the $ y $-axis has at least one corresponding value on the $ x $-axis

$ y = f(x) = x^4, f:\mathbb{R}\rightarrow\mathbb{R} $ Fig 1: $ f $ is not onto since there is no value on the $ x $-axis that can produce a negative real number on the $ y $-axis

One-to-One (Injective)

Definition: The function $ f $ is said to be injective (or to be one-one) if whenever $ x_1 $ is not equal to $ x_2 $, then $ f(x_1) $ is not equal to $ f(x_2) $. If $ f $ is an injective function, we also say that $ f $ is a injection.

In other words, if $ x_1 $ and $ x_2 $ are in the domain of $ f $ and if $ f $ is one-to-one if either of the following is true

$ \begin{align} &\bullet x_1 \neq x_2 \Rightarrow f(x_1) \neq f(x_2) \\ &\bullet f(x_1) = f(x_2) \Rightarrow x_1 = x_2 \end{align} $

e.g. Given that

$ f(x) = x^5, \forall x \in \mathbb{R}, f : \mathbb{R} \rightarrow \mathbb{R} $

then, $ f $ is onto the set R.

e.g. Given that

$ f(x) = x^2, \forall x \in \mathbb{R}, f : \mathbb{R} \rightarrow \mathbb{R} $

then, $ f $ is not onto the set R.

But if we define the domain of $ f $ such that

$ f(x) = x^2, \forall x \in \mathbb{R}, f : \mathbb{R} \rightarrow [0,\infty] $

then, $ f $ is not onto.

Bijection

Countability

para from bartle and sherbert on infinite sets denumerable or countable

illustrate with set of even numbers. and diagram irrational numbers diagonal method with diagram same cardinality

Cardinality

define cardinality

example

proof

References

• Bartle, Sherbert

• Kenney