- #1
evagelos
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How do we prove in propositional calculus :
...p^(qvr) <===> (p^q)v(p^r) semantically and syntactically
...p^(qvr) <===> (p^q)v(p^r) semantically and syntactically
Proof of p^(qvr) <=> (p^q)v(p^r)
- Thread starter evagelos
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In summary, there is a discussion about proving the equivalence of p^(qvr) and (p^q)v(p^r) in propositional calculus. The method of using truth tables is suggested, but there is uncertainty about whether this is allowed. Another potential method involving a contradiction is mentioned, but its effectiveness is uncertain. It is also mentioned that there may be a difference between a semantical proof and a syntactical proof. The conversation ends with a request for clarification and encouragement to share ideas.
- #2
CompuChip
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Well, what have you already come up with?
- #3
HallsofIvy
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Comparing truth tables will do it quickly and neatly. Are you not allowed to use that method?
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CompuChip
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I had to Google as well, as far as I http://www.rci.rutgers.edu/~cfs/472_html/Logic_KR/proplogic_proofs472.html , using truth tables would be the semantic proof.
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- #5
evagelos
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Semantical proof without using true tables ,i have one in mind but i am not very positive about it.Then syntactically how about a contradiction you think it could work ,although it looks a bit messy
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CompuChip
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It is not quite clear to me what you mean by a semantical proof, and a syntactical one.
Also, if you would post your idea we can have a look at it. Maybe you are on the right track but just need a last push, or maybe you even got it right but lack the confidence
Also, if you would post your idea we can have a look at it. Maybe you are on the right track but just need a last push, or maybe you even got it right but lack the confidence
1. What is the meaning of "Proof of p^(qvr) <=> (p^q)v(p^r)"?
The statement "Proof of p^(qvr) <=> (p^q)v(p^r)" is an equivalence in propositional logic, which means that the two statements on either side of the symbol <=> are logically equivalent. In other words, the truth value of one statement is always the same as the truth value of the other statement.
2. What is the purpose of proving p^(qvr) <=> (p^q)v(p^r)?
The purpose of proving this equivalence is to show that the two statements are interchangeable. This can be useful in simplifying complex logical expressions or solving logic problems.
3. How can I prove p^(qvr) <=> (p^q)v(p^r)?
In order to prove this equivalence, you can use logical equivalences and rules of inference to manipulate the expressions on either side of the <=> symbol until they are equivalent to each other.
4. Can you provide an example of how p^(qvr) <=> (p^q)v(p^r) can be used?
One example of how this equivalence can be used is in proving the distributive property of logical disjunction. By setting p to "It is raining", q to "I have an umbrella", and r to "I have a raincoat", we can show that the statement "If it is raining and I have an umbrella or I have a raincoat, then I am prepared for the rain" is equivalent to "If it is raining and I have an umbrella, or if it is raining and I have a raincoat, then I am prepared for the rain."
5. Are there any real-world applications of p^(qvr) <=> (p^q)v(p^r)?
Yes, this equivalence has applications in computer science, particularly in the design and analysis of logical circuits and algorithms. It is also used in artificial intelligence and natural language processing to simplify and manipulate logical expressions.
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