3.3: Formulas
- Page ID
- 99064
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(\subseteq\).