Proving the contrapositive

In summary, proof by contradiction and proof by contraposition are two different methods of proving a statement. Proof by contradiction assumes the opposite of the statement and derives a contradiction, while proof by contraposition assumes the negative of the conclusion and proves the negative of the hypothesis. Contraposition is a more constrained and constructive technique, while proof by contradiction is stronger and more general. Both methods can be used to prove a statement, but proof by contradiction may have more flexibility in choosing contradictory statements.
  • #1

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It can be proved by proof by contradiction. hence it is just a variant of it?
 
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  • #2
Not really. Say you have a statement p, a proof by contradiction would be not p implies false. Contrapositive is a direct proof. Say you have the statement p imples q. To prove this via contradiction you assume p is true and q is false then derive a contradiction, to prove this via contrapositive you assume not q is true (i.e. q is false) and prove that this imples p is false.
 
  • #3
Statement: p ⇒ q
Proof by contradiction: Assume p and ¬q; prove ⊥.
Proof by contraposition: Assume ¬q; prove ¬p.

A proof by contraposition can be turned into a proof by contradiction by adding the step "p ∧ ¬p ⇒ ⊥". But proofs by contradiction don't need to prove ¬p as an intermediate step; they can prove contradiction in other ways. So proof by contradiction is stronger.

On the other hand, the more constrained proof by contraposition is considered a constructive technique, while proof by contradiction is always considered nonconstructive. So if you have a choice use contraposition.
 
  • #4
If a statement is "P implies Q" then the contrapositive is "if not Q then not P" and is equivalent. Proving the contrapositive is a kind of proof by contradiction in which we assume the conclusion is false and arrive at a contradiction: "if not Q" is certainly assuming the conclusion is false and proving "not P" would give a contradiction with "if P".

However, "proof by contradiction" more general. You start with "if not Q" but only need to arrive at two statements that contradict one another. I have seen proofs in which the two contraditory statements have no apparent connection with the hypothesis, P.
 

What is the definition of proving the contrapositive?

Proving the contrapositive is a mathematical method used to prove the validity of a statement by showing that its logical equivalent, the contrapositive statement, is also true.

Why is proving the contrapositive important in mathematics?

Proving the contrapositive allows for more efficient and effective methods of proof. It also helps to clarify the reasoning behind a statement and can provide a stronger foundation for mathematical arguments.

What is the process for proving the contrapositive?

The process for proving the contrapositive involves first stating the original statement, then negating the conclusion and the hypothesis. Next, the negated statement is proven to be true using established mathematical principles. Finally, the contrapositive statement is concluded to be true based on the logical equivalence of the original and negated statements.

Can proving the contrapositive be used for all mathematical statements?

Yes, proving the contrapositive can be used for all mathematical statements. It is a valid method of proof that can be applied to a wide range of mathematical concepts and theories.

What are the benefits of using the contrapositive in mathematical proofs?

Using the contrapositive in mathematical proofs can help to simplify complex statements and make them easier to understand. It also allows for more efficient and elegant proofs, and can help to identify flaws or weaknesses in an argument.

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