Direct Proofs: Are They Just Introductions?

In summary, the conversation discussed the use of direct proofs in mathematics and questioned whether they are only used to introduce proofs or if they are also used to derive important theorems. The responder mentioned that many important theorems are indeed proved using direct proofs, such as the fundamental theorems of calculus. A link to a proof of the first part of the fundamental theorem of calculus was also provided.
  • #1
Benn
34
0
Hey guys,

I'm in a proof class right now. We've covered direct proofs and moved on, but I'm still curious about them. Is there any important theorem that has even been derived using a direct proof (assume p to show q) or are they mainly just used to introduce proofs? In class, we only ever cover proofs such as "if n ##\equiv## 1 (mod 2), then n2 ##\equiv## 1 (mod 8)." and the like.

Sorry, I can't get the tex to work out... aha, just got it working, nevermind
 
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  • #2
Benn said:
Is there any important theorem that has even been derived using a direct proof (assume p to show q) or are they mainly just used to introduce proofs?

Plenty of important theorems are proved using methods of direct proof. I presume that you are familiar with the fundamental theorems of calculus. The standard proofs of these results are done via direct proof. See here: http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Proof_of_the_first_part
 

FAQ: Direct Proofs: Are They Just Introductions?

What is a direct proof?

A direct proof is a method used in mathematics to prove that a statement is true by using logical steps and previously established knowledge. It involves starting with the given statement and using logical arguments to show that it must be true.

How does a direct proof differ from other proof methods?

A direct proof differs from other proof methods, such as indirect proof or proof by contradiction, in that it directly shows that the statement is true without assuming the opposite and proving its impossibility.

What are the basic steps of a direct proof?

The basic steps of a direct proof include: (1) stating the given information or hypothesis, (2) using definitions and previously established theorems to make logical deductions, (3) arriving at a conclusion that follows logically from the given information, and (4) writing a clear and concise proof that shows the logical steps taken.

What types of statements can be proved using direct proof?

Direct proof can be used to prove all types of statements, including mathematical theorems, logical statements, and even scientific theories. As long as the statement can be broken down into smaller logical steps and follows the rules of logic, it can be proved using direct proof.

What are the advantages of using direct proof?

Direct proof is a straightforward and systematic method of proving a statement, making it easy to follow and understand. It also relies on established knowledge and logical arguments, making it a reliable and widely accepted proof method in the scientific community.

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