List of the things I learnt today

  • Definition of a set
  • Different ways of representing a set
  • Different kinds of sets
  • Intervals
  • Combinations of sets
  • Set notation in functions

 

Definition of a set

A set is a collection of objects

Different ways of representing a set

Sets of objects are usually denoted by uppercase letters e.g.

let \mathit{A} be the set of all odd numbers.The objects contained in a set are called elements which are denoted by lowercase letters e.g. \mathit{a} is an element of \mathit{A}.

A set can be represented in two ways:

1.List

\mathit{A}=\{a, b, c, d\}

The set above is an finite list, you can count the number of elements.

\mathit{B}=\{a, b, c, d,... t\}

Is also a finite list. It’s important to present enough elements to produce a sequence if the use of ellipses is implemented for shorthand purposes.

\mathit{C}=\{1, 2, 3, 5, 8,... \}

This is an infinite set, you cannot count the number of elements in the set.

2. Set Builder Notation

\mathit{D}=\{e \in \mathbb{R} : \mathit{C} (e) \}

In the example above e is an element in the set \mathbb{R} where \mathit{C} (e) is a characteristic of the elements which are specific to set \mathit{D} or you can replace “:” with”|”. But it still carries the same meaning.

\mathit{D}=\{e \in \mathbb{R} | \mathit{C} (e) \}

moreover

\mathit{D}=\{e : \mathit{C} (e) \}

means e satisfies the characteristic \mathit{C} (e)

Different kinds of sets:

1.The empty set

The set containing no elements at all is denoted: \emptyset

2.Integers

Every number without a decimal or fractional part, denoted by \mathbb{Z}

3.Natural numbers

Represents all nonnegative integers: \mathbb{N}

4.Counting numbers

All positive integers: \mathbb{N}^{+}

5.Rational numbers

Every integer that can be written in the form: \frac{a}{b} where b \neq 0 denoted by \mathbb{Q}.

6.Real numbers

All the numbers on the number line, denoted by \mathbb{R}

7.Irrational numbers

Real numbers that cannot be written in the form \frac{a}{b} where b \neq 0 denoted by \mathbb{Q} ^ {\prime}

Intervals

let p and q belong to \mathbb{R}, then

(p,q) = \{ y \in \mathbb{R} : p < y < q\}

[p,q) = \{ y \in \mathbb{R} : p \leq y < q\}

(p,q] = \{ y \in \mathbb{R} : p < y \leq q\}

[p,q] = \{ y \in \mathbb{R} : p \leq y \leq q\}

(p,\infty) = \{ y \in \mathbb{R} : y > p\}

(- \infty,p) = \{ y \in \mathbb{R} : y < p \}

[p, \infty) = \{ y \in \mathbb{R} : y \geq p\}

(- \infty,p] = \{ y \in \mathbb{R} : y \leq p\}

(- \infty, \infty) = \mathbb{R}

Combining sets

\mathit{A} \cup \mathit{B}

is all the elements in \mathit{A} or \mathit{B}

\mathit{A} \cap \mathit{B}

is all the elements in \mathit{A} and \mathit{B}

if

\mathit{A} \cap \mathit{B} = \emptyset

then \mathit{A} and \mathit{B} are called a disjoint.

Set notation in functions

\mathit{F} : \mathit{D} \to \mathbb{R}

means that every element in the set \mathit{D} is assigned to an element in the set \mathbb{R} by the function \mathit{F}.

My first time using LaTeX. And I survived the fires of UCT.

How clear is this post?