A bijective function is both injective and surjective.

A function f is

injective, if for every y, there existsat most one(≤ 1) x such that f(x) = y.

In other words, for any y in the codomain, there could either be a unique x such that f(x) = y, or there could be none.

A function f is

surjective, if for every y, there existat least one(≥ 1) x such that f(x) = y.

In other words, for any y in the codomain, there could be x₁, x₂, x₃,… such that f(x₁) = f(x₂) = f(x₃)…

The classic PageRank algorithm from Google was something that they used to rank the importance of websites.

So the idea is this: Imagine that I am a Google user, and I want to look something up on the search engine (say *Stephen Hawking*). There will be millions of websites on the Internet that contain the keyword *Stephen Hawking* in it. How does Google know which of them is the most relevant, or how does it know which website to put at the top of the search results?

Given E **⊂** X where E and X are both sets, we call E to be **dense** in X, when every point of X is either in E, or is a limit point of E.

Every real number in ℝ is either a rational number, or is a limit point of the rational numbers ℚ.

Why? Because in any ε-neighborhood of a real number x, you can always find a q ∈ ℚ such that x < q< x + ε. In other words, the open ball Br(x, ε) always have a non-empty intersection with the rational numbers.

This intuitively…

Groups, rings and fields are mathematical objects that share a lot of things in common. You can always find a ring in a field, and you can always find a group in a ring.

A group is a set of symbols {…} with a law ✶defined on it. Every symbol has an inverse 1/x , and a group has an identity symbol 1.

More formally, a group (G, ✶) satisfies following axioms:

- Closure: ✶ : G × G → G
- Associativity: (a ✶ b) ✶ c = a ✶ (b ✶ c), for any a, b, c ∈ G.
- …

First, let’s assume that I have an operation **a ⊙ b**, where **a** and **b** are two random symbols, and **⊙ **is an arbitrary operation. In order to make **a ⊙ b **have a valid answer (let’s say, **a ⊙ b = c **is a valid construction), we need to show that **⊙ **is a** well-defined **operation.

For any well-defined operations, if I take a value **a’** that is equal to **a**, and a value **b’** that is equal to **b**, then **a ⊙ b = a’ ⊙ b’ = c.**

Note that **a’ **and **a **might not look exactly…

To make a group from scratch, I need to start with a set of symbols.

I want to make my own set, so I picked 3 random symbols Ε, Ο, Δ to form set, and I will call this set **G**.

Notice that there is nothing special in the choices I made. You could have picked an apple, a banana, and a cherry for your set.

Now, if we just leave the set as how it is, it will be pretty boring set because there’s nothing we can learn from it. To make this set more interesting, I want to…

I am a student studying pure mathematics.