Disjunction elimination: Difference between revisions
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imported>Soreningraham m Added a hatnote linking to Disjunctive syllogism since the shared name could bring readers to the wrong article. |
imported>Semaurer01 make display of expression more clear. |
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| field = [[Propositional calculus]] | | field = [[Propositional calculus]] | ||
| statement = If a statement <math>P</math> implies a statement <math>Q</math> and a statement <math>R</math> also implies <math>Q</math>, then if either <math>P</math> or <math>R</math> is true, then <math>Q</math> has to be true. | | statement = If a statement <math>P</math> implies a statement <math>Q</math> and a statement <math>R</math> also implies <math>Q</math>, then if either <math>P</math> or <math>R</math> is true, then <math>Q</math> has to be true. | ||
| symbolic statement = <math>\ | | symbolic statement = <math> | ||
\begin{aligned} | |||
1.\quad & P \to Q \\ | |||
2.\quad & R \to Q \\ | |||
3.\quad & P \lor R \\ | |||
\therefore\quad & Q | |||
\end{aligned} | |||
</math> | |||
}} | }} | ||
{{Transformation rules}} | {{Transformation rules}} | ||
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An example in [[English language|English]]: | An example in [[English language|English]]: | ||
:If I'm inside, I have my wallet on me. | :1. If I'm inside, I have my wallet on me. | ||
:If I'm outside, I have my wallet on me. | :2. If I'm outside, I have my wallet on me. | ||
:It is true that either I'm inside or I'm outside. | :3. It is true that either I'm inside or I'm outside. | ||
:Therefore, I have my wallet on me. | :Therefore, I have my wallet on me. | ||
It is the rule can be stated as: | It is the rule can be stated as: | ||
:<math>\ | :<math> | ||
\begin{aligned} | |||
1.\quad & P \to Q \\ | |||
2.\quad & R \to Q \\ | |||
3.\quad & P \lor R \\ | |||
\therefore\quad & Q | |||
\end{aligned} | |||
</math> | |||
where the rule is that whenever instances of "<math>P \to Q</math>", and "<math>R \to Q</math>" and "<math>P \lor R</math>" appear on lines of a proof, "<math>Q</math>" can be placed on a subsequent line. | where the rule is that whenever instances of "<math>P \to Q</math>", and "<math>R \to Q</math>" and "<math>P \lor R</math>" appear on lines of a proof, "<math>Q</math>" can be placed on a subsequent line. | ||