Proof By Contradiction Exercises

In summary, the person is looking for exercises to practice mathematical proof by contradiction. They have completed their A-levels in Math but did not cover much on proof. They are currently reading a book on mathematical proof but it focuses on logic and they are looking for something more similar to their A-level material. They are seeking suggestions or advice on finding exercises without having to learn basic logic theory first. Two recommended books by the person are "The Nuts and Bolts of Proofs" and "How to Read and Do Proofs."
  • #1
Galadirith
109
0
Hi guys, I am looking to find some exercises, preferably online to practice Mathematical Proof by Contradiction.

I have just finished my A-levels in the UK doing Maths and Further Maths and very little is done in the way of mathematical proof. The is only a single chapter on proof by induction, with a short bit explaining the concept of Proof by induction and then many questions asking you to proove by induction. The question ask either to proove that and inequality is correct, proove that the solution to a reccurence relation is correct or prove that the general formula to a series is correct.

Now logic in the sense of propositional logic or otherwise actually is not discussed at all or even required as a prerequisit to doing this (true mathematical logic isn't disscused at all in the maths A-levels in the UK), everything is done in "Plain english" as it were :D. And I am really looking for a similar thing to practice proof by contradiction. I am currently reading "100% mathematical proof", which is really enjoyable, but proof is approched from a completely logic standpoint with the first half of the book really didicated to familiraising the reader with propositional and predicate logic, and in truth I know that is the way it should be done, other wise youll end up with unrigourous proofs, which arnt really proofs then. And it seems that all the material I can find that includes exercises for contradiction take the completely logical standpoint, presented in with logic maths text or philosiphy texts.

Can anyone suggest something similar to what I have found in my A-level for induction, without all the logic shrouding it, I am sure most would advise against it but I really have thought about it and would love to find some exercises that I can do without the need to learn basic logic theory first. Any help, suggestions or advise would be really appreciated, thanks a lot guys :D
 
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  • #2
Very late reply, but in case someone else is interested, here are a couple books I find useful:
"The Nuts and Bolts of Proofs," by Antonella Cupillari, published by Wadsworth Publishing Co.
ISBN 0-534-10320-0

"How to Read and Do Proofs," by Daniel Solow, published by John Wiley and Sons
ISBN 0-471-51004-1
 

Related to Proof By Contradiction Exercises

1. What is a proof by contradiction exercise?

A proof by contradiction exercise is a type of mathematical or logical proof where the validity of a statement is shown by assuming the opposite and then showing that it leads to a contradiction, thus proving that the original statement must be true.

2. How do you approach a proof by contradiction exercise?

To approach a proof by contradiction exercise, you first assume that the statement is false and then use logical steps to derive a contradiction. This usually involves using known facts and properties to reach a contradiction, such as the law of non-contradiction.

3. What is the difference between a direct proof and a proof by contradiction?

A direct proof is a straightforward, step-by-step argument that directly proves the statement, while a proof by contradiction assumes the opposite of the statement and then shows that it leads to a contradiction, thus indirectly proving the original statement.

4. When is it useful to use a proof by contradiction?

A proof by contradiction is useful when a direct proof is difficult or impossible to construct, or when the statement to be proved is of the form "if p, then q" and it's easier to prove "not q" assuming "p" is true, and then using a proof by contradiction to show that "not q" leads to a contradiction.

5. Are all mathematical or logical statements provable by contradiction?

No, not all statements can be proved by contradiction. Some statements may be true but cannot be proven using a contradiction. Additionally, some statements may be true but their negation cannot be proven by contradiction. It is important to carefully consider the statement and the available tools and techniques before deciding to use a proof by contradiction.

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