157 lines
4.8 KiB
TeX
157 lines
4.8 KiB
TeX
\section{Structures}
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\definition{}
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A \textit{universe} is a set of meaningless objects. Here are a few examples:
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\begin{itemize}
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\item $\{a, b, ..., z\}$
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\item $\{0, 1\}$
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\item $\mathbb{Z}$, $\mathbb{R}$, etc.
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\end{itemize}
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\definition{}
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A \textit{structure} consists of a universe and a set of \textit{symbols}. \par
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A structure's symbols give meaning to the objects in its universe.
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\vspace{2mm}
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Symbols come in three types:
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\begin{itemize}
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\item \textit{Constant symbols}, which let us specify specific elements of our universe. \par
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Examples: $0, 1, \frac{1}{2}, \pi$
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\vspace{2mm}
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\item \textit{Function symbols}, which let us navigate between elements of our universe. \par
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Examples: $+, \times, \sin{x}, \sqrt{x}$ \par
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\note{In this handout, symbols we usually call \say{operators} are also called functions. \par
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The only difference between $a + b$ and $+(a, b)$ is notation.}
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\vspace{2mm}
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\item \textit{Relation symbols}, which let us compare elements of our universe. \par
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Examples: $<, >, \leq, \geq$ \par
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\vspace{2mm}
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\end{itemize}
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The equality check $=$ is \textbf{not} a relation symbol. It is included in every structure by default. \par
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By definition, $a = b$ is true if and only if $a$ and $b$ are the same element of our universe.
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\vspace{3mm}
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\example{}
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The first structure we'll look at is the following:
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$$
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\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)
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$$
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\vspace{2mm}
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This is a structure over the universe $\mathbb{Z}$ that provides the following symbols:
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\begin{itemize}
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\item Constants: \tab $\{0, 1\}$
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\item Functions: \tab $\{+, -\}$
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\item Relations: \tab $\{<\}$
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\end{itemize}
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\vspace{2mm}
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If we look at our set of constant symbols, we see that the only integers
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we can directly refer to in this structure are 0 and 1. If we want any
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others, we must define them using the tools this structure offers.
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\vspace{2mm}
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To \say{define} an element of a set, we need to write a sentence that is only true for that element. \par
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For example, if we want to define 2 in the structure above, \par
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we could use the sentence \say{$2$ is the $x$ that satisfies $[1 + 1 = x]$.} \par
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This is a valid definition because $2$ is the \textbf{only} element of $\mathbb{Z}$ for which $[1 + 1 = x]$
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evaluates to \texttt{true}.
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\problem{}
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Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)$.
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\begin{solution}
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The sentences \say{$x$ where $[x + 1 = 0]$} and \say{$x$ where $[0 - 1 = x]$} both work.
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\end{solution}
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\vfill
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\pagebreak
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Let us formalize what we found in the previous two problems. \par
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\definition{Formulas}
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A \textit{formula} in a structure $S$ is a well-formed string
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of constants, functions, relations, \par and logical operators.
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\vspace{2mm}
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You already know what a \say{well-formed} string is: $1 + 1$ is fine, $\sqrt{+}$ is nonsense. \par
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For the sake of time, I will not provide a formal definition. It isn't particularly interesting.
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\vspace{2mm}
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As a quick example, the formula $\phi(x) = [1 + 1 = x]$ evaluates to \texttt{true} when $x$ is 2 \par
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and to \texttt{false} otherwise.
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\definition{Free Variables}
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A formula can contain one or more \textit{free variables.} These are denoted $\varphi{(a, b, ...)}$. \par
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Formulas with free variables let us define \say{properties} that certain objects have. \par
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For example, $x$ is a free variable in the formula above. \par
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$\varphi(2)$ is \texttt{true} and $\varphi(-3)$ is \texttt{false}. \par
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\vspace{2mm}
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This \say{free variable} notation is much like the function notation we are used to: \par
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$\varphi(x) = [x > 0]$ is similar to $f(x) = x + 1$, since the values of $\varphi(x)$ and $f(x)$ depend on $x$.
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\definition{Definable Elements}
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Say $S$ is a structure with a universe $U$. \par
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We say an element $e \in U$ is \textit{definable in $S$} if we can write a formula that only $e$ satisfies.
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\problem{}
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Define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bigr)$. \par
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\hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$. Also, $2 \times 2 = 4$.}
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\begin{solution}
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$2$ is the only element in $\mathbb{Z}^+$ that satisfies $\varphi(x) = [x \times x = 4]$.
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\end{solution}
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\vfill
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\problem{}
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Try to define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr)$. \par
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Why can't you do it?
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\begin{solution}
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This isn't possible. We could try $\varphi(x) = [x \times x = 4]$, but this is satisfied by both $2$ and $-2$. \par
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We have no way to distinguish between negative and positive numbers.
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\begin{instructornote}
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Actually, it is. Bonus problem: how? \par
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Do this after understanding quantifiers.
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\end{instructornote}
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\end{solution}
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\vfill
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\problem{}
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What numbers are definable in the structure $\Bigl( \mathbb{R}^+_0 ~\big|~ \{1, 2, \div \} \Bigr)$?
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\begin{solution}
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We can define powers of two, positive and negative.
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If you're clever, you can define many more: $\sqrt{2}, \sqrt[3]{2}, ...$.
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\end{solution}
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\vfill
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\pagebreak |