Page 23 - 'Blast_Into_Math

Blast into Math! Pure mathematics: the proof of the pudding is in the eating

• Contrapositive. Like a proof by contradiction, you should always begin a proof by
contrapositive by stating that your proof is by contrapositive. Next you assume that the
conclusion is not true. Then you use the assumption that the conclusion is not true to
make true statements until you have shown that the hypothesis cannot be true. You will
have therefore shown that if the conclusion is not true, then the hypothesis cannot be true,
which is precisely the contrapositive. By the Contrapositive Proposition you have proven the
original statement.

If you continue your mathematical endeavors, you will inevitably face a mathematical statement that
you want to prove, but you don’t know how. (If you don’t believe me, then go prove – or disprove – the
Riemann Hypothesis.) It is an exciting and intimidating challenge! But, when you have no idea where
to start, what do you do? How do you start a proof?

1. Read the proof of a simple theorem. Try to understand the main ideas and the main reasons

the theorem is true.
2. Attempt to prove a simple statement. Start small, and you will have better luck and more fun
building up to bigger theorems.
3. To begin your proof review the definitions of everything in the statement of the hypothesis
and conclusion. Start thinking about what the definitions imply. Do some of these
implications get you closer to the conclusion? In your first few proofs in this book you
should be able to reach the conclusion in just a few steps using only the definitions.
4. If the definitions and their implications do not show you a direct logical path to the
conclusion, try working backwards. Think about the conclusion and what true statements,

closer to the hypothesis, would imply the conclusion. If you can do this, you can continue
taking logical steps backwards until you reach the hypothesis. Then, you can try to use these
steps to construct a proof which begins at the hypothesis and reaches the conclusion.
5. If these direct approaches are not working, try assuming the conclusion is false. Think about
what this would imply and try to either prove that something impossible happens (proof by
contradiction) or prove that the hypothesis must be false (proof by contrapositive).
6. If you are still stuck, turn the statement into specific examples. Work out these examples

and see how in those specific cases the statement is true. Try to find a pattern among the
examples and generalize this to a logical argument which proves the statement in all cases.
7. If you are still stuck, think about similar statements and theorems. How are they proven?
Can you do something similar in your proof?
8. Finally, it is unlikely but possible that what you are trying to prove is false. Try to
determine if the statement is true and experiment with examples to see if you can find a
counterexample which proves that the statement is actually false!

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