Bfh-sts at 13:31, 20 October 2025

2025-10-20T13:31:42Z

← Older revision		Revision as of 15:31, 20 October 2025
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	= ~~Absorption~~ laws =		= Neutral and dominance laws =

	The ~~absorption~~ laws ~~show~~ how ~~certain combinations of~~ conjunction and disjunction ~~can be simplified by "absorbing" one proposition into another~~.		The neutral and dominance laws describe how conjunction and disjunction interact with the constants ''true'' and ''false''.

	== ~~Statements~~ ==		== Neutral laws ==
	* ~~p ∨ (~~p ∧ q) ≡ p		* p ∧ wahr ≡ p
	* p ~~∧ (~~p ∨ q) ≡ p		* p ∨ falsch ≡ p

			== Dominance laws ==
			* p ∨ wahr ≡ wahr
			* p ∧ falsch ≡ falsch

	== Explanation ==		== Explanation ==
	~~Adding extra conditions~~ that ~~are already implied by~~ ''p'' ~~does not change~~ the ~~truth value~~.		* The neutral laws state that combining with the neutral element leaves the proposition unchanged.
	~~These laws allow expressions to be reduced in complexity~~.		– 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 OR (~~I study AND ~~I rest)~~" ~~is logically equivalent to~~ "I study".		* "I study AND true" ≡ "I study".
	* "I study ~~AND (~~I study ~~OR I rest)~~" ~~is logically equivalent to~~ "I study".		* "I study OR false" ≡ "I study".
			* "I study OR true" ≡ true.
	~~== Truth Table (First Law) ==~~		* "I study AND false" ≡ false.
	~~{\| class=~~"~~wikitable~~"
	~~! p !! q !! p ∧ q !! p ∨ (p ∧ q)~~
	\|-
	~~\| T \|\| T \|\| T \|\| T~~
	\|-
	~~\| T \|\| F \|\| F \|\| T~~
	\|-
	~~\| F \|\| T \|\| F \|\| F~~
	\|-
	~~\| F \|\| F \|\| F \|\| F~~
	\|}

	== Truth ~~Table (Second Law)~~ ==		== Truth Tables ==
	{\| class="wikitable"		{\| class="wikitable"
	! p !! q !! p ∨ q !! p ∧ ~~(p ∨ q)~~		! p !! p ∧ wahr !! p ∨ falsch !! p ∨ wahr !! p ∧ falsch
	\|-
	~~\| T \|\| T \|\| T \|\| T~~
	\|-
	~~\| T \|\| F \|\| T \|\| T~~
	\|-		\|-
	\| F \|\| T \|\| T \|\| F		\| 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]]

Bfh-sts: 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..."

2025-10-20T13:31:33Z

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..."

New page

= 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 equivalent to "I study".
* "I study AND (I study OR I rest)" is logically equivalent to "I study".

== Truth Table (First Law) ==
{| class="wikitable"
! p !! q !! p ∧ q !! p ∨ (p ∧ q)
|-
| T || T || T || T
|-
| T || F || F || T
|-
| F || T || F || F
|-
| F || F || F || F
|}

== Truth Table (Second Law) ==
{| class="wikitable"
! p !! q !! p ∨ q !! p ∧ (p ∨ q)
|-
| T || T || T || T
|-
| T || F || T || T
|-
| F || T || T || F
|-
| F || F || F || F
|}

[[Category:Propositional logic (Aussagenlogik)]]
[[Category: Diskrete Mathematik I (BZG1155pa) 25/26]]

Neutral and dominance laws - Revision history

Bfh-sts at 13:31, 20 October 2025