- #1

Mr Davis 97

- 1,462

- 44

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- I
- Thread starter Mr Davis 97
- Start date

- #1

Mr Davis 97

- 1,462

- 44

- #2

mfb

Mentor

- 36,291

- 13,366

That should be ##\neg (p \rightarrow q) = \neg q \wedge p##. If you are careful with statements like "for all" and "there exists", then they are all the same thing.##\neg (p \rightarrow q) = \neg p \wedge q##

- #3

- 4,065

- 1,645

Subject to that, the two approaches are logically equivalent in classical first-order predicate logic, which is all that mathematicians that don't specialise in logic worry about.

In intuitionist logic and other logics where some of the basic axioms such as ##\neg\neg p\leftrightarrow p## are not accepted, the approaches may give different results.

- #4

Edgardo

- 705

- 15

To prove [itex]p \rightarrow q[/itex]:

- In proof by contraposition you start by assuming that [itex]\neg q[/itex] is true and derive the statement [itex]\neg p[/itex]. Here, the path is clear, i.e. you start at [itex]\neg q[/itex] and arrive at [itex]\neg p[/itex].

- In proof by contradiction your start by assuming that the opposite of [itex]p \rightarrow q[/itex] is true. So you assume that [itex]p \wedge \neg q[/itex] is true and derive some contradiction. Here the path is not clear, nobody is going to tell you what the contradiction is and what it looks like.

- #5

Svein

Science Advisor

- 2,274

- 785

Another point against "proof by contradiction" is that it does not help you in solving anything, it just says that the assumption is proved (but not how).

Share:

Changing the Statement
Combinatorial proofs & Contraposition

- Last Post
- Math Proof Training and Practice

- Replies
- 5

- Views
- 499

- Last Post

- Replies
- 8

- Views
- 581

- Last Post

- Replies
- 22

- Views
- 541

- Replies
- 3

- Views
- 454

- Last Post

- Replies
- 7

- Views
- 608

- Last Post
- Math Proof Training and Practice

- Replies
- 4

- Views
- 507

- Last Post

- Replies
- 5

- Views
- 653

- Last Post

- Replies
- 5

- Views
- 520

- Last Post

- Replies
- 3

- Views
- 456

- Last Post

- Replies
- 5

- Views
- 792