Notes taken for CSCB36 course at UofT, this post is for Chapter 0, mainly talks about sets and fundamental mathematical units.

## Sets

If an object is $a$ in $A$, then we write $a \in A$, if not, we write $a \notin A$.

If a set has 0 elements, we call that an empty set, or $\emptyset$.

Number of elements within a set $A$ is called its size or cardinality, denoted by $|A|$.

• For infinite set $A$, $|A| = \infty$
• For empty set $\emptyset$, $|\emptyset| = 0$

### Describing Sets

There are 2 ways to describe sets.

• Extensional: List all elements within a set. For example $\{1,2,3\}$.
• Intensional: Describle the set with format $\{x: \text{x is odd number}\}$.

### Relationship Between Sets

• $A \subseteq B$: If every element in $A$ is also in $B$, then $A$ is a subset of $B$.
• $B \supseteq A$: If $A \subseteq B$ then this holds.
• $A \subseteq B$ and $B \subseteq A$ then $A = B$.
• $A \subseteq B$ and $A \neq B$ then $A$ is a proper subset of $B$, or $A \subset B$.
• $B \supset A$ if $A \subset B$.

Interesting properties:

$\emptyset \subseteq A$ for all set $A$.

$\emptyset \subset A$ for all set $A \neq \emptyset$.

### Set Operations

• Union of $A$ and $B$ or $A \cup B$, is the set of elements that belongs to $A$ or $B$.

• Intersection of $A$ and $B$ or $A \cap B$, is the set of element that belongs to $A$ and $B$. If $A \cap B = \emptyset$, then $A$ and $B$ are disjoint.

• Difference of $A$ and $B$ or $A - B$, is the set of elements that belong to $A$ but not to $B$.