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.Empty set
\[\mathit{E}=\{\}\]
or
\[\mathit{E} = \phi\]
2.Integer
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
\[\mathit{A} \cap \mathit{B} = \phi\]
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.
Try not putting in a return before and after all your Latex symbols. Can’t you just enclose them in [latex] some Latex code [/latex] within the same paragraph?
I tried doing that at first but then it didn’t work at all. Right now what I do is use this format $latex(code)$ and when I do that the latex code doesn’t fit in with the rest of the text. So I resorted to returning before and after each latex entry. I’ll try that method again though. Your input is appreciated.