Introduction to Sets

Definition of Set

 

def : A set is a collections of things that are referred to as the elements of the set

A = {1, 2, 4, 5, a, ... }

A : name of set (Capital Letter)

{ } : braces (set brackets)

1, 2, 4, 5, a : elements (Small Letter)

... : ellipsis

 

if $A = \{a, b, c\}$

$a \in A$

a is in set A

$d \notin A$

d is not in set A

 

 

Number Sets

 

$\mathbb{N}$ : Natural Numbers
$\{1, 2, 3, 4, \cdots\}$

$\mathbb{Z}$ : Integers
$\{\cdots, -2, -1, 0, 1, 2, \cdots\}$

$\mathbb{Q}$ : Rational Numbers
$e.g.\ \frac{4}{2}, \ -\frac{3}{5}$

$\bar{\mathbb{Q}}$ : Irrational Numbers
$e.g.\ \pi, e, \sqrt{2}$

$\mathbb{R}$ : Real numbers = both Rational and Irrational Numbers

$\mathbb{I}$ : Imaginary Numbers
$e.g.\ i = \sqrt{-1}$

$\mathbb{C}$ : Complex Numbers = combination of Real and Imaginary number (in the form of $a + bi$)
$e.g.\ -3 + 4i$

 

 

Set Equality

When are two sets equal?

Axiom of Extension : a set is determined by what its elements are - not the order in which they might be listed or the fact that some elements might be listed more than once

 

set doesn't care about order and duplication of elements at all

 

 

Set Builder Notation

(조건 제시법)

$A = \{1, 4, 7, \cdots\}$

is the same with

$\{x \in \mathbb{N}\ \vert\ x = 3k - 2, k \in \mathbb{N} \}$