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 $B$.

Arbitrary Number of Operations