Laws of Algebra of Propositions
Below mentioned are the laws of Algebra of Propositions
Idempotent Law
- p ∨ p ≅ p
- p ∧ p ≅ p
The truth table of conjunction and disjunction of a proposition with itself will equal the proposition.
Associative Law
- (p ∨ q) ∨ r ≅ p ∨ (q ∨ r)
- (p ∧ q) ∧ r ≅ p ∧ (q ∧ r)
Associative Law states that propositions also follow associativity and can be written as mentioned above.
Distributive Law
- p ∨ (q ∧ r) ≅ (p ∨ q) ∧ (p ∨ r)
- p ∧ (q ∨ r) ≅ (p ∧ q) ∨ (p ∧ r)
Distributive Law states that propositions also follow the distribution and can be written as mentioned above.
Commutative Law
- p ∨ q ≅ q ∨ p
- p ∧ q ≅ q ∧ p
It states that propositions follow commutative property i.e if a=b then b=a
Identity Law
- p ∨ T ≅ T
- p ∨ F ≅ p
- p ∧ T ≅ p
- p ∧ F ≅ F
where T is a Tautology, F is a Contradiction and p is a proposition.
De Morgan’s Law
In propositional logic and boolean algebra, De Morgan’s laws are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation. In formal language, the rules are written as –
- [Tex]\neg (p\wedge q) \equiv \neg p \vee \neg q [/Tex]
- [Tex]\neg (p\vee q) \equiv \neg p \wedge \neg q [/Tex]
Proof by Truth Table
[Tex]\begin{tabular}{ ||c||c||c||c||c||c||c||c||c||c|| } \hline p & q & \neg p & \neg q & p\wedge q & \neg p\vee \neg q & p\vee q & \neg p\wedge \neg q \\ \hline T & T & F & F & T & F & T & F \\ \hline T & F & F & T & F & T & T & F \\ \hline F & T & T & F & F & T & T & F \\ \hline F & F & T & T & F & T & F & T \\ \hline\end{tabular}[/Tex]
Involution Law
- ~~p ≅ p
Complement Law
- p ∨ ~p ≅ T
- p ∧ ~p ≅ F
- ~T ≅ F
- ~F ≅ T
where T is a Tautology, F is a Contradiction and p is a proposition.
Propositional Logic – Set 2
This article explores fundamental laws and concepts in the algebra of propositions like Idempotent, Associative, Distributive, and Commutative Laws, as well as special conditional statements.
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