Proof of (p ⇒ q) =(¬p ∨ q) in Mathematical Logic

Thread starter Byeonggon Lee
Start date Apr 2, 2015
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Logic Mathematical logic Proof

In summary, the conversation discusses the meaning and use of the law of contrapositive, a proof technique in logic. It states that if p implies q, then the negation of p must imply the negation of q. The steps involved in a "(p ⇒ q) =(¬p ∨ q) proof" include assuming the negation of the conclusion, using logical steps to show that the negation of the premise must also be true, and ultimately proving the validity of the original statement. However, this technique can only be used to prove statements in the form of "p ⇒ q" and there are other proof techniques such as proof by contradiction and proof by cases.

Apr 2, 2015

Byeonggon Lee

I am studying mathematical logic by a pdf file. But there is no proof about this therorem so I don't understand.. How to prove this?

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Apr 2, 2015

WWGD

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https://www.physicsforums.com/threads/transformation-rules-in-boolean-algebra.806074/#post-5060328

Apr 2, 2015

Byeonggon Lee

WWGD said:

https://www.physicsforums.com/threads/transformation-rules-in-boolean-algebra.806074/#post-5060328

Truth table aha! thanks

Related to Proof of (p ⇒ q) =(¬p ∨ q) in Mathematical Logic

1. What is the meaning of "(p ⇒ q) =(¬p ∨ q) proof"?

The statement "(p ⇒ q) =(¬p ∨ q) proof" is a logical expression that represents a proof technique known as the law of contrapositive. It states that if p implies q, then the negation of p must imply the negation of q. In other words, if p is true, then q must also be true, and if p is false, then q must also be false.

2. How is the law of contrapositive used in a proof?

The law of contrapositive is used in a proof by showing that if the negation of the conclusion is true, then the negation of the premise must also be true. This can be done by assuming the negation of the conclusion, and then using logical steps to show that the negation of the premise must also be true. This ultimately proves that the original statement is valid.

3. What are the steps involved in a "(p ⇒ q) =(¬p ∨ q) proof"?

The steps involved in a "(p ⇒ q) =(¬p ∨ q) proof" are as follows:

Assume the negation of the conclusion, ¬q.
Use logical steps to show that ¬p must also be true.
Rewrite the original statement using the negation of the premise and the negation of the conclusion.
Show that the new statement is logically equivalent to the original statement.
Conclude that the original statement is true.

4. Can the law of contrapositive be used to prove any statement?

No, the law of contrapositive can only be used to prove statements that are in the form of "p ⇒ q" (if p then q). It cannot be used to prove statements that are not in this form, such as "p ∧ q" (p and q) or "p ∨ q" (p or q).

5. Are there any other proof techniques that are similar to the law of contrapositive?

Yes, the law of contrapositive is just one of many proof techniques in logic. Other techniques include proof by contradiction, proof by cases, and proof by mathematical induction. These techniques all involve using logical steps to show that a statement is true or false.