Monoxdifly said:\(\displaystyle p\wedge\neg (q\vee r)\)
=\(\displaystyle p\wedge(\neg q\wedge\neg r)\)
=\(\displaystyle (p\wedge\neg q)\wedge (p\wedge\neg r)\)
So, it implies 2)
solakis said:Sorry,but i am trying to find out how you got \(\displaystyle (p\wedge\neg q)\wedge (p\wedge\neg r)\) from
\(\displaystyle p\wedge(\neg q\wedge\neg r)\)
Olinguito said:And ($\wedge$) is distributive over itself since it is associative, commutative and idempotent: https://proofwiki.org/wiki/Associative_Commutative_Idempotent_Operation_is_Distributive_over_Itself.
Aren’t the expressions in red equal?solakis said:(a*b)*(a*c)= a*[b*(a*c)]
Now where wiki gets : (a*b)*(a*c)=a*(b*a)*c ??
Olinguito said:Aren’t the expressions in red equal?
solakis said:By using associativity you can get them equal.
But i am asking ,how wiki gets, a*(b*a)*c from (a*b)*(a*c) by ‹using associativity
Parenthesis are very important in this kind of proof otherwise a lot of wrong proofs can be given
Look at the following proof:
(a*b)*(a*c)= (a*a)*b*c=............By commutativity
=a*b*c.................By indepondency
I like Serena said:If we do not have associativity, parentheses are indeed very important.
If we do have associativity, parentheses are redundant, and often left out.
In this case we have associativity, don't we?
solakis said:Can you give an example of a system with a binary opperation where by using associativity you can prove that parenthesis are redundant
Monoxdifly said:\(\displaystyle p\wedge\neg (q\vee r)\)
=\(\displaystyle p\wedge(\neg q\wedge\neg r)\)
=\(\displaystyle (p\wedge\neg q)\wedge (p\wedge\neg r)\)
So, it implies 2)
| p | q | r | ( | p | & | ~ | ( | q | ∨ | r | ) | ) | → | ( | ( | p | & | ~ | q | ) | ∨ | ( | p | & | ~ | r | ) | ) | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| T | T | T | T | F | F | T | T | T | T | T | F | F | T | F | T | F | F | T | ||||||||||||||
| T | T | F | T | F | F | T | T | F | T | T | F | F | T | T | T | T | T | F | ||||||||||||||
| T | F | T | T | F | F | F | T | T | T | T | T | T | F | T | T | F | F | T | ||||||||||||||
| T | F | F | T | T | T | F | F | F | T | T | T | T | F | T | T | T | T | F | ||||||||||||||
| F | T | T | F | F | F | T | T | T | T | F | F | F | T | F | F | F | F | T | ||||||||||||||
| F | T | F | F | F | F | T | T | F | T | F | F | F | T | F | F | F | T | F | ||||||||||||||
| F | F | T | F | F | F | F | T | T | T | F | F | T | F | F | F | F | F | T |
Propositional calculus is a type of mathematical logic that deals with the relationships between propositions or statements. It is a formal system that uses symbols and rules to determine the validity of logical arguments.
The basic components of propositional calculus include propositions (statements that can be either true or false), logical connectives (such as "and", "or", and "not"), and inference rules (used to derive new propositions from existing ones).
Propositional calculus is used in computer science to create logical reasoning algorithms that can be implemented in computer programs. It is also used in artificial intelligence, natural language processing, and automated theorem proving.
One limitation of propositional calculus is that it cannot handle the concept of quantification, which is necessary for dealing with statements involving "all" or "some" objects. It also does not account for the ambiguity or context of natural language.
Propositional calculus has various real-world applications, such as in cryptography, circuit design, and database query optimization. It is also used in legal reasoning and philosophy to analyze arguments and determine their validity.