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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 2: $ 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} $

You may recall the informal method of checking whether functions are one-to-one using the horizontal line test. This is illustrated in the following examples in figures three and four.

$ y = f(x) = x, f:\mathbb{R}\rightarrow\mathbb{R} $ Fig 3: Any arbitrary horizontal line $ y = c $ intersects the function's graph only once so we can infer that $ f $ is injective.

$ y = f(x) = \sin(x), f:\mathbb{R}\rightarrow\mathbb{R} $ Fig 3: The $ x $-axis intersects the graph of the function multiple times. It does not pass the horizontal line test. Therefore we can conclude that $ f $ is not injective.

Bijection

Definition: If $ f $ is both injective and surjective, then $ f $ is said to be bijective. If $ f $ is bijective, we also say that $ f $ is a bijection

References

• R. G. Bartle, D. R. Sherbert, "Preliminaries" in "An Introduction to Real Analysis", Third Edition, John Wiley and Sons Inc. 2000 ch. 1, sect. 1.1, pp 18.

• R. Kenney. MA 301. "An Introduction to Proof through Real Analysis", Lecture Notes. Faculty of the Department of Mathematics, Purdue University, Spring 2012

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