MHB How to give a proof of tautologies?

[Henry R]

Okay. Hello =) =) I am confuse regarding to this matter.

Now, I'm going to write about tautologies.

A proposition p is always true is called a tautology. A proposition p that is always false is called a contradiction.

Example :
p v p is an example of tautology
P ^ P is an example of contradiction

Suppose that the compound proposition p is made up of
propositions p 1 ... p n and compound proposition q is made up of propositions q 1 ... q n , we say that p and q are logically equivalent and write it as p ≡ q
provided that given any truth values of p 1 ... p n and truth values of q 1 ... q n , either p and q are both true or p and q are both false.

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Here's the question : Which of the following are tautologies? If the statement is a tautology, give a proof using the appropriate rules of logic. (Avoid using truth tables if possible.) If it is not a tautology, then justify your answer by giving an appropriate example for the following questions below :

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Thank you so much for reading my thread. =)

 

[Evgeny.Makarov]

p v p is an example of tautology
P ^ P is an example of contradiction

These must mean $p\lor\bar{p}$ and $p\land\bar{p}$.

Here's the question : Which of the following are tautologies?

Have you figured this out using truth tables or common sense? For the first formula, is it true that $p$ always implies $p$ or $q$? For the second one, suppose that if you are free, then you go to the movies, and if you are busy, you also go to the movies. If it is known that you are either free, busy or went to the movies, does it follow that you are watching a movie?

If the statement is a tautology, give a proof using the appropriate rules of logic.

The set of appropriate rules of logic differs from one textbook or course to the next. It would be nice if you listed them.

 

[Henry R]

These must mean $p\lor\bar{p}$ and $p\land\bar{p}$.

Have you figured this out using truth tables or common sense? For the first formula, is it true that $p$ always implies $p$ or $q$? For the second one, suppose that if you are free, then you go to the movies, and if you are busy, you also go to the movies. If it is known that you are either free, busy or went to the movies, does it follow that you are watching a movie?

The set of appropriate rules of logic differs from one textbook or course to the next. It would be nice if you listed them.

Um just wondering? Can you see the pictures?? the png pictures? It just I can't see it. But, by the way thank you so much.

 

[Evgeny.Makarov]

Um just wondering? Can you see the pictures?? the png pictures?

Do you mean the three images in post #1? Yes, I see them.

It just I can't see it.

That's strange. Maybe you can start a thread about in the http://mathhelpboards.com/questions-comments-feedback-25/ subforum. You could post a screenshot there of how you see post #1.

 

[Henry R]

Do you mean the three images in post #1? Yes, I see them.

That's strange. Maybe you can start a thread about in the http://mathhelpboards.com/questions-comments-feedback-25/ subforum. You could post a screenshot there of how you see post #1.

Oh ya, now I can see it if I'm online.