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.