Definitions

Natural Numbers: $\N = \{0,1,\dots\}$

Power Set: $\mathcal{P}(S)=\{A: A\sub S \}$

• Example: $\mathcal{P}(\{1, 2\})=\{\{\}, \{1\}, \{2\}, \{1, 2\}\}$

Functions

Function: $f:A\to B$

• A: Domain (Set of all inputs) B: Codomain (Set of all possible outputs)
• All $a \in A$ is mapped to some value in $B$ Not all $b \in B$ is mapped.
• Criteria: $\forall a \in A, f(a) \in B$
• Example

Injective / One-to-One: $f:A\to B$ is injective if

• $\forall x, y \in A, (x \neq y \implies f(x) \neq f(y))$ ”If the inputs are different, the outputs are different”
• Alternative: $\forall x, y \in A, (f(x) = f(y) \implies x = y)$
• Pigeonhole Principle: $|B| < |A| \Leftrightarrow \neg\exists f:A\to B$
• Example: Hilbert’s Hotel - Injective function fitting infinite number of people (domain) to infinite number of rooms (codomain)

Surjective / Onto: $f:A \to B$ is surjective if

• $\forall b \in B, \exists a \in A \text{ s.t. } f(a) = b$ ”Every element of the codomain is mapped to”
• Note: Always specify domain and codomain when defining a function

Bijective: Both injective and surjective

Cantor’s Theorem