Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision | |||
|
sav08:propositional_logic_informally [2012/02/21 15:09] vkuncak |
sav08:propositional_logic_informally [2015/04/21 17:30] (current) |
||
|---|---|---|---|
| Line 27: | Line 27: | ||
| === Negation (logical 'not') === | === Negation (logical 'not') === | ||
| - | \[\begin{array}{c|c} | + | \begin{equation*}\begin{array}{c|c} |
| p & \lnot p \\\hline | p & \lnot p \\\hline | ||
| {\sl false} & {\sl true} \\ \hline | {\sl false} & {\sl true} \\ \hline | ||
| {\sl true} & {\sl false} | {\sl true} & {\sl false} | ||
| \end{array} | \end{array} | ||
| - | \] | + | \end{equation*} |
| === Conjunction (logical 'and') === | === Conjunction (logical 'and') === | ||
| - | \[\begin{array}{c|cc} | + | \begin{equation*}\begin{array}{c|cc} |
| \land & {\sl false} & {\sl true} \\ \hline | \land & {\sl false} & {\sl true} \\ \hline | ||
| {\sl false} & {\sl false} & {\sl false} \\ \hline | {\sl false} & {\sl false} & {\sl false} \\ \hline | ||
| {\sl true} & {\sl false} & {\sl true} | {\sl true} & {\sl false} & {\sl true} | ||
| \end{array} | \end{array} | ||
| - | \] | + | \end{equation*} |
| === Disjunction (logical 'or') === | === Disjunction (logical 'or') === | ||
| - | \[\begin{array}{c|cc} | + | \begin{equation*}\begin{array}{c|cc} |
| \lor & {\sl false} & {\sl true} \\ \hline | \lor & {\sl false} & {\sl true} \\ \hline | ||
| {\sl false} & {\sl false} & {\sl true} \\ \hline | {\sl false} & {\sl false} & {\sl true} \\ \hline | ||
| {\sl true} & {\sl true} & {\sl true} | {\sl true} & {\sl true} & {\sl true} | ||
| \end{array} | \end{array} | ||
| - | \] | + | \end{equation*} |
| === Implication ('if') === | === Implication ('if') === | ||
| - | \[\begin{array}{c|cc} | + | \begin{equation*}\begin{array}{c|cc} |
| \rightarrow & {\sl false} & {\sl true} \\ \hline | \rightarrow & {\sl false} & {\sl true} \\ \hline | ||
| {\sl false} & {\sl true} & {\sl true} \\ \hline | {\sl false} & {\sl true} & {\sl true} \\ \hline | ||
| {\sl true} & {\sl false} & {\sl true} | {\sl true} & {\sl false} & {\sl true} | ||
| \end{array} | \end{array} | ||
| - | \] | + | \end{equation*} |
| Check validity of these implications: | Check validity of these implications: | ||
| Line 69: | Line 69: | ||
| === Logical Eqivalence ('if and only if') === | === Logical Eqivalence ('if and only if') === | ||
| - | \[\begin{array}{c|cc} | + | \begin{equation*}\begin{array}{c|cc} |
| \leftrightarrow & {\sl false} & {\sl true} \\ \hline | \leftrightarrow & {\sl false} & {\sl true} \\ \hline | ||
| {\sl false} & {\sl true} & {\sl false} \\ \hline | {\sl false} & {\sl true} & {\sl false} \\ \hline | ||
| {\sl true} & {\sl false} & {\sl true} | {\sl true} & {\sl false} & {\sl true} | ||
| \end{array} | \end{array} | ||
| - | \] | + | \end{equation*} |
| ===== Evaluating Propositional Formulas ===== | ===== Evaluating Propositional Formulas ===== | ||
| Line 81: | Line 81: | ||
| Let us evaluate formula | Let us evaluate formula | ||
| - | \[ | + | \begin{equation*} |
| ((p \rightarrow q) \land\ ((\lnot p) \rightarrow r)) \leftrightarrow ((p \land q)\ \lor\ ((\lnot p) \land r)) | ((p \rightarrow q) \land\ ((\lnot p) \rightarrow r)) \leftrightarrow ((p \land q)\ \lor\ ((\lnot p) \land r)) | ||
| - | \] | + | \end{equation*} |
| for all values of its parameters. Let us draw formula as a tree. We introduce a column for each tree node. We obtain the value of a tree node by looking at the value of its children and applying the truth table for the operation in the node. The root gives the truth value of the formula. | for all values of its parameters. Let us draw formula as a tree. We introduce a column for each tree node. We obtain the value of a tree node by looking at the value of its children and applying the truth table for the operation in the node. The root gives the truth value of the formula. | ||
| Line 109: | Line 109: | ||
| Here is a small list of tautologies. | Here is a small list of tautologies. | ||
| - | \[\begin{array}{l} | + | \begin{equation*}\begin{array}{l} |
| (p \rightarrow q) \leftrightarrow ((\lnot p) \lor q) \\ | (p \rightarrow q) \leftrightarrow ((\lnot p) \lor q) \\ | ||
| (p \leftrightarrow q) \leftrightarrow ((p \rightarrow q) \land (q \rightarrow p)) \\ | (p \leftrightarrow q) \leftrightarrow ((p \rightarrow q) \land (q \rightarrow p)) \\ | ||
| Line 126: | Line 126: | ||
| ((p \lor q) \rightarrow r) \leftrightarrow ((p \rightarrow r) \land (q \rightarrow r)) \\ | ((p \lor q) \rightarrow r) \leftrightarrow ((p \rightarrow r) \land (q \rightarrow r)) \\ | ||
| ((p \rightarrow {\sl false}) \leftrightarrow (\lnot p) | ((p \rightarrow {\sl false}) \leftrightarrow (\lnot p) | ||
| - | \end{array}\] | + | \end{array}\end{equation*} |
| Suggest another tautology. | Suggest another tautology. | ||