Sets
| Name |
Set notation |
Logic operator |
Description |
| Subset |
$S\subseteq T$ |
$\forall x(x\in S\to x\in T)$ |
TRUE if every element of $S$ is in $T$ |
| Superset |
$S\supseteq T$ |
$\forall x(x\in T\to x\in S)$ |
TRUE if every element of $T$ is in $S$ |
| Set equality |
$S = T$ |
$\forall x(x\in S\leftrightarrow x\in T)$ |
- |
| Union |
$S\cup T$ |
$\{x |
x\in S \vee x\in T\}$ |
| Intersection |
$S\cap T$ |
$\{x |
x\in S \wedge x\in T\}$ |
| Difference |
$S-T$ |
$\{x |
x\in S \vee x\notin T\}$ |
Sequence
- A sequence is an ordered collection of elements
- $a=(a_1,a_2,a_3,...,a_n)$
- $n$ notates the length of the sequence
- $a_i$ denotes the i-th element of the sequence
Sequence properties
- increasing: $\forall i\ a_i\le a_{i+1}$
- decreasing: $\forall i\ a_i\ge a_{i+1}$
- Subsequence:
- Original: A = (1, 2, 3, 4, 5)
- Subsequence or A: (1, 3, 5)
Matrices
$$
A=\begin{bmatrix} a_{1,1} &a_{1,2} &\cdots &a_{1,n}\\ a_{2,1} &a_{2,2} &\cdots &a_{2,n} \\ \vdots &\vdots & &\vdots \\ a_{m,1}&a_{m,2} &\cdots & a_{m,n}\end{bmatrix}
$$
<aside>
💡 Exam tips:
Write like this for procedure points
</aside>
Function
- Given two sets: domain $X$, range $Y$
- A function $f:X\to Y$ maps each element of X to an element of Y
- defined as explicit set of pairs
- e.g., $\{ (1,3),(2,6),(3,9),\cdots\}$
- Alternatively written as, $f(x)=y$, where $x$ is an element of $X$ and $y$ is an element of $Y$
- defined by an equation
- e.g., $f(x)=3x$
Types of function
- One-to-one function: All values of $x$ maps to unique value $y$
- Onto function: Some values of $x$ maps to same value of $y$
Relations