Neutral and dominance laws: Difference between revisions
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(Created page with "= Absorption laws = The absorption laws show how certain combinations of conjunction and disjunction can be simplified by "absorbing" one proposition into another. == Statements == * p ∨ (p ∧ q) ≡ p * p ∧ (p ∨ q) ≡ p == Explanation == Adding extra conditions that are already implied by ''p'' does not change the truth value. These laws allow expressions to be reduced in complexity. == Examples == * "I study OR (I study AND I rest)" is logically equivalen...") |
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= | = Neutral and dominance laws = | ||
The | The neutral and dominance laws describe how conjunction and disjunction interact with the constants ''true'' and ''false''. | ||
== | == Neutral laws == | ||
* | * p ∧ wahr ≡ p | ||
* p | * p ∨ falsch ≡ p | ||
== Dominance laws == | |||
* p ∨ wahr ≡ wahr | |||
* p ∧ falsch ≡ falsch | |||
== Explanation == | == 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 == | == Examples == | ||
* " | * "I study AND true" ≡ "I study". | ||
* "I study | * "I study OR false" ≡ "I study". | ||
* "I study OR true" ≡ true. | |||
* "I study AND false" ≡ false. | |||
== Truth | == Truth Tables == | ||
{| class="wikitable" | {| class="wikitable" | ||
! p !! | ! p !! p ∧ wahr !! p ∨ falsch !! p ∨ wahr !! p ∧ falsch | ||
|- | |- | ||
| | | T || T || T || T || F | ||
|- | |- | ||
| F || F || F || F | | F || F || F || T || F | ||
|} | |} | ||
[[Category:Propositional logic (Aussagenlogik)]] | [[Category:Propositional logic (Aussagenlogik)]] | ||
[[Category: Diskrete Mathematik I (BZG1155pa) 25/26]] | [[Category: Diskrete Mathematik I (BZG1155pa) 25/26]] | ||
Latest revision as of 14:31, 20 October 2025
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 |