102 lines
2.2 KiB
TeX
102 lines
2.2 KiB
TeX
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\section{Logical Algebra}
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\definition{}
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Odds are, you are familiar with \textit{logical symbols}. \par
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In this handout, we'll use the following:
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\begin{itemize}
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\item $\lnot$: not
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\item $\land$: and
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\item $\lor$: or
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\item $\rightarrow$: implies
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\item $()$, parenthesis.
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\end{itemize}
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The function of these is defined by \textit{truth tables}:
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\begin{center}
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\begin{tabular}{ c | c | c }
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\multicolumn{3}{ c }{and} \\
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\hline
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$A$ & $B$ & $A \land B$ \\
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\hline
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F & F & F \\
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F & T & F \\
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T & F & F \\
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T & T & T
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\end{tabular}
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\hfill
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\begin{tabular}{ c | c | c }
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\multicolumn{3}{ c }{or} \\
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\hline
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$A$ & $B$ & $A \lor B$ \\
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\hline
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F & F & F \\
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F & T & T \\
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T & F & T \\
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T & T & T
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\end{tabular}
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\hfill
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\begin{tabular}{ c | c | c }
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\multicolumn{3}{ c }{implies} \\
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\hline
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$A$ & $B$ & $A \rightarrow B$ \\
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\hline
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F & F & T \\
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F & T & T \\
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T & F & F \\
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T & T & T
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\end{tabular}
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\hfill
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\begin{tabular}{ c | c }
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\multicolumn{2}{ c }{not} \\
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\hline
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$A$ & $\lnot A$ \\
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\hline
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T & F \\
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F & T \\
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~ & ~ \\
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~ & ~ \\
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\end{tabular}
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\end{center}
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\vspace{2mm}
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$A \land B$ is only true if both $A$ and $B$ are true. $A \lor B$ is only true if $A$ or $B$ (or both) are true. \par
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$\lnot A$ is the opposite of $A$, which is why it looks like a \say{negative} sign. \par
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\vspace{2mm}
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$A \rightarrow B$ is a bit harder to understand. Read aloud, this is \say{$A$ implies $B$.} \par
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The only time $\rightarrow$ is false is when $T \rightarrow F$. Think about it: why does this make sense? \par
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\problem{}
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Evaluate the following.
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\begin{itemize}
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\item $(T \land F) \lor T$
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\item $(\lnot (F \lor \lnot T) ) \rightarrow T$
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\item $A \rightarrow T$ for any $A$
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\item $(\lnot (A \rightarrow B)) \rightarrow A$ for any $A,B$
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\end{itemize}
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\vfill
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\pagebreak
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\problem{}
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Show that $\lnot (A \rightarrow \lnot B)$ is equivalent to $A \land B$. \par
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\hint{Use a truth table}
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\vfill
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\problem{}
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Can you express $A \lor B$ using only $\lnot$, $\rightarrow$, and $()$?
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\begin{solution}
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$((\lnot A) \rightarrow B)$
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\end{solution}
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\vfill
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Note that both $\land$ and $\lor$ can be defined using the other logical symbols. \par
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The only logical symbols we \textit{need} are $\lnot$, $\rightarrow$, and $()$. \par
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We include $\land$ and $\lor$ to simplify our logical expressions.
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\pagebreak
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