Neutral and dominance laws
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Neutral and dominance laws
The neutral and dominance laws describe how conjunction and disjunction interact with the constants true and false.
Neutral laws
- p ∧ wahr ≡ p
- p ∨ falsch ≡ p
Dominance laws
- p ∨ wahr ≡ wahr
- p ∧ falsch ≡ falsch
Explanation
- The neutral laws state that combining with the neutral element leaves the proposition unchanged.
– In conjunction, true is neutral. – In disjunction, false is neutral.
- The dominance laws state that combining with the dominant element forces the result.
– In conjunction, false dominates. – In disjunction, true dominates.
Examples
- "I study AND true" ≡ "I study".
- "I study OR false" ≡ "I study".
- "I study OR true" ≡ true.
- "I study AND false" ≡ false.
Truth Tables
| p | p ∧ wahr | p ∨ falsch | p ∨ wahr | p ∧ falsch |
|---|---|---|---|---|
| T | T | T | T | F |
| F | F | F | T | F |