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I'm a first year Physics student and I have some trouble approaching Proofs by Contradiction in some of my Math classes. Once I get the first 2 or 3 statements I can finish the proof but a lot of the time I can't immediately get the first step. Any tips to approach this better (some examples would be helpful)?

PSPACEhard
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Sreekar Voleti
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    Could you please provide an example? – PSPACEhard May 26 '15 at 08:10
  • For example - The proof that there are infinite number of primes (Euclid) – Sreekar Voleti May 26 '15 at 08:13
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    @DoYouEvenScienceBro: This proof is not (necessarily) a proof by contradiction. – Regret May 26 '15 at 08:16
  • Yes but I'm trying to gain an intuition of how to use that method specifically so I'd like to use that in this case – Sreekar Voleti May 26 '15 at 08:18
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    There are plenty of examples on this side (for example listed on the right hand sided under "Related") which might give you some tips. Such as http://math.stackexchange.com/questions/451328/a-proof-by-contradiction – Henry May 26 '15 at 08:21
  • In addition, I'll say many so-called ‘proofs by contradiction’ are not really so. Infinity of primes (in Euclid's proof) is a typical example: indeed it is proved that any finite list of primes is incomplete. – Bernard May 26 '15 at 08:23

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There are infinite number of primes.

The proof by contradiction is as such:

Suppose to the contrary there exists a finite number of primes.

Let's call this set of finite number of primes

$$p_1\text{,...,}p_n$$

If you add 1 to this set, we have a new set and we shall denote this new set as S:

$$S=\left\{p_1\text{,..., }p_n\text{$\}$+1}\right.$$

There are thus 2 cases:

This new set could be prime or it could not be prime.

Case 1: This set is prime

If this set is prime, then we are done for we have shown that there exists another prime number outside the original finite set.

Case 2: This set is not prime

If this set is not prime, then

$$\left.\text{S-$\{$}p_1\text{,..., }p_n\right\}$$ can be divided by some prime.

The difference after subtraction is 1.

But recall that the only integer that can divide 1 is 1 but 1 is not a prime number. So we conclude that only initial assumption of there being finitely many primes as false.

Mathematicing
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  • What does $\left{p_1\text{,..., }p_n\text{$}$+1}\right.$ mean? I know you mean that you add $1$ to the product of primes $p_1 p_2 \cdots p_n$, but I've never seen addition being performed between a set and a real number – Ovi Aug 28 '16 at 13:55
  • Also, see: https://math.stackexchange.com/questions/30127/is-there-an-intuitionist-i-e-constructive-proof-of-the-infinitude-of-primes – aeyalcinoglu May 17 '17 at 14:57