- $(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )$
- Order of quantifiers matters.
## Relations
- Cartesian product:
- $A \times B = \{ (p, q) : p \in A \land q \in B \}$
- Def: A relation $R$ on a set $\mathbb X$ is a subset of $\mathbb X \times \mathbb X$ ($R \subseteq \mathbb X \times \mathbb X$)
- Graph of a function $f()$: $\{ (x, f(x) : x \in Dom(f) \}$
- Properties of:
- Reflexivity: $(\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x$
- Symmetricity: $[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$
- Transitivity: $(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$
- Antisymmetricity: $(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$
- Equivalence relations:
- Def: $R \subseteq \mathbb X \times \mathbb X$ is said to be an equivalence relation *iff* $R$ is reflexive, symmetric and transitive.
- Congruence modulo n: $p R q \equiv n | p - q$
- Def R - and equivalence relation of $\mathbb X$: The _equivalence class_ of an element $x \in \mathbb X$ is the set $[x]_R = \{ y \in \mathbb X : x R y \}$
- Every $x \in \mathbb X$ belongs to the equivalence class of some element $a$.
- A partition is a set containing subsets of some set $\mathbb X$ such that their collective symmetric difference equals $\mathbb X$. A partition of is a set $\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}$ such that:
- Def: A partial order $R$ on $\mathbb X$ is said to be *"total"* iff $(\forall x, y \in \mathbb X) (x R y \lor y R x)$
- Def: A subset $B$ of $\mathbb X$ is called a chain *"chain"* iff $B$ is totally ordered by $R$
- $C(\mathbb X)$ - the set of all chains in $(\mathbb X, R)$
- A chain $D$ in $(\mathbb X, R)$ is called a maximal chain iff $D$ is a maximal element in $(C(\mathbb X), R)$
- $K \subseteq \mathbb X$ is called an antichain in $(\mathbb X, R)$ iff $(\forall p, q \in K) (p R q \lor q R p \implies p = q)$
- Def: $R$ is a partial order on $\mathbb X$, $R$ is called a *well* order iff $R$ is a total order on $X$ and every nonempty subset $A$ of $\mathbb X$ has the smallest element
## Induction
- If $\phi$ is a propositional function defined on $\mathbb N$, if:
- $\phi(1)$
- $(\forall n \geq 1) \phi(n) \implies \phi(n+1)$
- $(\forall k \geq 1) \phi(k)$
## Functions
- $f: \mathbb X \to \mathbb Y$
- Def: $f \subseteq \mathbb X \times \mathbb Y$ is said to be a function if:
- $(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$
- $(\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)$
- Types of functions $f: \mathbb X \to \mathbb Y$:
- $f$ is said to be an injection ( 1 to 1 function) iff $(\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$
- $f$ is said to be a surjection (onto function) iff $(\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y$
- If $f^{-1}$ is a function from $\mathbb Y \to \mathbb X$ then $f^{-1}$ is called the inverse function for $f$
- Fact: $f^{-1}$ is a function iff $f$ is a *bijection* (1 to 1 and onto)
- For some set $\mathbb A$ the image of $\mathbb A$ by $f$ is $f(\mathbb A) = \{ f(x) : x \in \mathbb A \}$. We can also define the inverse of an image even when the function itself isn't invertible: $f^{-1}(\mathbb A)$
