Neutral and dominance laws

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