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Logic and Proof

Logic and Proof

Postulates and Theorems

Typically, the things in your reasons column will be one of the following: Given, some property of equality, a geometrical postulate (or axiom), or a theorem proved earlier.

We already know what "given" and the properties of equality are. What about the other stuff?

Postulates (also called axioms) are running assumptions about the objects we are talking about. For example, in geometry, we assume that exactly one line can be drawn between any two points. Obviously, whether that is true depends upon how one precisely defines the terms "line" and "point."

Rather than being bogged down by overcomplicated definitions, it's more convenient to just say, "Look, we all accept that there's only one line through these two points, so let's feel free to use it."

We all agree that there are 24 hours in a day. We could argue endlessly about how a day should be defined (when does a rotation of the earth begin and end?) and how hours came to exist, but all this bickering doesn't get us to school on time. At some point we simply declared that time is measured in hours after midnight, and that there are 24 hours in a day, and that's that.

Unfortunately, we got a bit carried away with these postulates and now there are time zones and daylight savings time, which we also have to deal with. Still, you can say, "Let's catch a 7:00 movie," and others will understand you. That's because they share the same common assumptions (or postulates) about how time is measured.

In Euclid's famous Elements, an example of ancient Greek geometry, he lists at the beginning 23 definitions and only 5 postulates. Instead of listing postulates all at once, we will introduce them when they naturally come up. Also, be warned: there will be significantly more than 5.

Finally, a theorem is some fact that's already been proved. Since we've already shown it to be true, we can use it to show other things are true. This can be especially convenient when using the theorem saves you a lot of writing.

We can use this idea to build even longer trains.

Sample Problem

Given: A = B, B = C, C = D, D = E.
Prove: A = E

StatementsReasons
1. A = BGiven
2. B = CGiven
3. C = DGiven
4. D = EGiven
5. A = DTransitive Property (1, 2, and 3)
6. A = ETransitive Property (5 and 4)

Of course, theorems and postulates can be used in all kinds of proofs, not just formal ones. Paragraph or informal proofs lay out a logical argument in paragraph form, while indirect proofs assume the reverse of the given hypothesis to prove the desired conclusion. Proofs are like a bag of Bertie Bott's Every Flavor Beans: they might come in different (and sometimes awful-tasting) flavors, but they're all made of the same basic stuff. Except for that proofs are made of fact while Bertie Bott's are made of magic.

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