1Something about boolean algebras¶
If $B$ is a finite boolean algebra and $A = \{a_1, \ldots, a_n\}$ is the set of atoms of $B$, then the map taking $b$ to $\{x \in A, x \subseteq b\}$ is an isomorphism of partial orders.
That is, $f$ is a bijection between $B$ to $\mathcal{P}(A) (f(\bot) = \varnothing)$ and for $b, b' \in B, b \leq b' \iff f(b) \subseteq f(b')$
2Predicate logic¶
2.1Logical symbols¶
 Connectives ($\lnot, \land, \lor, \to, \iff$ as before)
 Variables ($x,y,x', \bar{y}, z, \ldots$)
 Brackets ($(,)$)
 Quantifiers ($\exists, \forall$)
 The equality symbol ($=$)
2.2Signatures¶
A signature is a collection of nonlogical symbols, each with an assigned kind and arity^{1}.
2.2.1Types of symbols¶
 Constant symbols ($c, d', \ldots$)
 For each $n$, an $n$ary function symbol that stands for a function $\mathcal{M}^n \to \mathcal{M}$ where $\mathcal{M}$ is part of the underlying set. The arity is part of the signature. ^{not really sure what this is saying}
 For each $n$, an $n$ary predicate symbol $P,Q \ldots$ (stand for subsets of $\mathcal{M}^n$
Note: We treat these nonlogical symbols as prefix operators. For instance, we would write $+x3$, not $x+3$.
2.2.2An example¶
Suppose your structure is the Euclidean plane.
"$x$ is between $y$ and $z$
$Bxyz$ for some predicate symbol $B$
Note: the signature may be empty or have just one of the above kinds or two or all three!
2.2.3Terms¶
Given a signature $\Sigma$, a term is one of the following:
 Any variable or function symbol
 for any $n$nary function symbol $f$ and term $t_1, \ldots t_n$ we have a term $ft_1, \ldots, f_n$
The standard signature for studying $\mathbb{N}$ has $\{0, S, +, \cdot, <\}$, where:
 $0$ is a constant
 $S$ is a unary function symbol (the successor function; informally, $Si = (i + 1)$)
 $+, \cdot$ are binary function symbols
 $<$ is a binary predicate symbol

The number of inputs for a function. Unary, binary, ternary, etc. ↩