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All swans on the moon are green!

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Yes, it’s true. Well, mathematically anyway.

Let’s use a bit of logic and reasoning. Consider the statement ‘all swans on the moon are green’. How do we prove this? There are two mathematically equivalent ways of approaching this. One is obviously to collect together all swans on the moon and then show that they are all green. There are a few practical issues with this, so lets consider an alternative.

First let me take you through some logic. To prove that a statement is true, you can prove that it is ‘not false’. For example, the statement ‘all rabbits are animals’ is the same as ‘there does not exist a rabbit which is not an animal’. With me?

So, the statement ‘all swans on the moon are green’ is the same as the statement ‘you can’t find a swan on the moon which isn’t green’. Let’s work with the second statement. If the second statement is true, then so is the first statement. Let’s go again with the same logic.What if the second statement is false? This would mean that ‘you can find a swan on the moon which isn’t green’. Can you? Can you fetch me/take a picture for me/prove in some way that there is a swan on the moon that isn’t green? Nope, you can’t (admittedly because – as far as I know – there are no swans on the moon!). That means our assumption that the statement was false, is false – in other words, the statement is true.

If you’re with me, then what we’ve done is prove that ‘you can’t find a swan on the moon which isn’t green’, which means that ‘all swans on the moon are green’. Note that it doesn’t say anything about the size of the set…

This can all be written out using set notation and quantifiers, so if you’re that way inclined, why not try and write it out. If not, at least it’s something you can tell your friends!

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