3.3: Formulas
Loosely speaking, a formula is a mathematical expression with variables. Corresponding to each variable, \(x_{i}\) , appearing in a formula is a universe, \(U_{i}\) , from which that variable may be substituted.
DEFINITION. Open formula An open mathematical formula in variables \(x_{1}, \ldots, x_{n}\) is a mathematical expression in which substitution of the \(x_{i}(1 \leq i \leq n)\) by specific elements from \(U_{i}\) yields a mathematical statement.
EXAMPLE 3.8. Consider the formula, \[x^{2}+y^{2}=z^{2}\] in variables \(x, y\) and \(z\) , all with universe \(\mathbb{N}\) . Any substitution of the variables with natural numbers results in a statement. For instance, \[3^{2}+4^{2}=5^{2}\] or \[1^{2}+1^{2}=2^{2} .\] Of course, statements can be true or false, so some substitutions yield true statements, while others will yield false statements.
In discussing a general formula in \(n\) variables, we may use the notation \(P\left(x_{1}, \ldots, x_{n}\right)\) . For \(1 \leq i \leq n\) , let \(U_{i}\) be the universe of the variable \(x_{i}\) , and \(a_{i} \in U_{i}\) . The statement that results from the substitution of \(a_{i}\) for \(x_{i}, 1 \leq i \leq n\) , is written \(P\left(a_{1}, \ldots, a_{n}\right)\) .
If \(P\left(x_{1}, \ldots, x_{n}\right)\) is a formula in variables \(x_{1}, \ldots, x_{n}\) , and for \(1 \leq i \leq\) \(n, U_{i}\) is the universe of \(x_{i}\) , then we may think of \(\left(x_{1}, \ldots, x_{n}\right)\) as a single variable with universe \(U=\prod_{1 \leq i \leq n} U_{i}\) .
Formulas can fulfill many purposes in mathematics:
(1) Characterize relationships between quantities
(2) Define computations (3) Define sets
(4) Define functions.
EXAMPLE 3.9. Consider an open formula, \(P(x, y)\) , in two variables, \[x^{2}+y^{2}=1,\] with universe \(\mathbb{R}^{2}\) . That is, the universe of \(x\) is \(\mathbb{R}\) and the universe of \(y\) is \(\mathbb{R}\) . One way to think of \(P(x, y)\) is as a means to partition \(\mathbb{R}^{2}\) into two sets:
(1) the subset of the Cartesian Plane for which the equation is true, namely the unit circle;
(2) the subset of the Cartesian Plane for which the equation is false, the complement of the unit circle in \(\mathbb{R}^{2}\) .
DEFINITION. Characteristic set, \(\chi_{P}\) Let \(P(x)\) be a formula, and \(U\) the universe of the variable \(x\) . The subset of \(U\) for which the formula \(P\) holds is written \(\chi_{P}\) . The set \(\chi_{P}\) is called the characteristic set of \(P(x)\) .
So, \[\chi_{\neg P}=U \backslash \chi_{P} .\]
3.3.1. Formulas and Propositional Connectives.
Propositional logic is easily extended to formulas. Let \(P(x)\) and \(Q(x)\) be formulas in the variable \(x\) , with universe \(U\) . Let \[R(x)=P(x) \wedge Q(x) .\] Then the characteristic set of \(R(x)\) is given by \[\chi_{R}=\{a \in U \mid T(P(a) \wedge Q(a))=1\}\] Hence \[\chi_{R}=\chi_{P} \cap \chi_{Q} .\] The propositional connective \(\wedge\) is strongly associated with the set operation \(\cap\) . Similarly \(\vee\) may be associated with \(\cup, \neg\) with complement (in \(U\) ), and \(\Rightarrow\) with \(\subseteq\) .