Proof by Contrapositive

In summary, the speaker has started university and has a lot of assignments, including a proof question involving inequalities. They have successfully completed the first part using induction, but are struggling with the second part and are considering using the contrapositive method. They have researched and believe they understand the process, but are seeking confirmation.
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2^Oscar
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Hi guys,

I've just started university this week and I've been given a mountain of assignments. One of them has a proof question in it. Since this is an assignment I want to make clear that I don't want help with the actual proof.

In the first part of the question I'm asked to, given a particular inequality (lets call it A), show that another inequality (this one B) is true. This was a trivial proof by induction.

Next I am asked to prove that given inequality B, that inequality A is true. In the workbook it mentions in passing something called the contrapositive which is something I haven't encountered before. I can't get the answer to drop out using the standard induction method so I assume I need to use this new one.

My understanding of the contrapositive from looking at articles on the internet is that, to show A is true given B, I need to contradict the inequality of A (for example if A has < I need to flip it to [tex]\geq[/tex]) and then prove, using basically the same induction process as the first part of the question, that the contradiction of B is true (e.g. if B had sign < I need to show through induction that the same inequality just with a [tex]\geq[/tex] sign is true). I then would state that it is true for the contrapositive hence the statement is true.

I guess my question is; have I correctly understood the process of proof by contrapositive outlined above?

Thanks for the help in advance,
Oscar
 
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  • #2
Yes, the "contrapositive" of the statement "If B then A" is "if not A then not B" so you would prove "If A then B", using the contrapositive" by starting "if A is not true, then ..." and using that to prove that B is not true.
 

1. What is "Proof by Contrapositive"?

Proof by Contrapositive is a method of mathematical proof that involves showing that if the negation of a statement is true, then the original statement must also be true. This is done by assuming the negation of the statement and using logical deductions to arrive at a contradiction, thereby proving the original statement to be true.

2. How is "Proof by Contrapositive" different from other proof methods?

Proof by Contrapositive is different from other proof methods, such as direct proof or proof by contradiction, because it involves proving a statement by showing that its negation leads to a contradiction. This method is particularly useful when the original statement is difficult to prove directly.

3. What are the steps involved in "Proof by Contrapositive"?

The steps involved in "Proof by Contrapositive" are as follows:

  1. Assume the negation of the statement to be proved.
  2. Use logical deductions to arrive at a contradiction.
  3. Since a contradiction has been reached, the original statement must be true.

4. When is "Proof by Contrapositive" useful?

"Proof by Contrapositive" is useful when the original statement is difficult to prove directly, or when the negation of the statement is easier to work with. It is also commonly used in proofs involving implications or conditional statements.

5. Can "Proof by Contrapositive" be used for all mathematical statements?

Yes, "Proof by Contrapositive" can be used for all mathematical statements, as long as the statement can be written in the form of an implication or conditional statement. However, in some cases, other proof methods may be more suitable or efficient.

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