Logic and Proof
Introduction to :
So how do we really know when two things are equal? We rely primarily upon properties, and we aren't talking real estate. We mean the properties of reflexivity, symmetry, and transitivity. These play a special role in geometry, so they are extra-important to remember.
The reflexive property states that A = A. That's some deep stuff, man.
Think of the reflexive property as the reflexible property. Take a contortionist, for example. He'll be a contortionist if he's standing up or if he's sitting on his own head. It doesn't matter how flexible he is, he'll still be a contortionist.
The reflexive property may seem obvious (and it is) but it's still useful, especially when dealing with geometrical figures. And it's way easier than folding yourself into a ball.
The symmetric property states that if A = B, then B = A. (Did you notice that it's a conditional statement, too?) This is mainly useful for reorganizing our expressions on the page, since it lets us flip statements across the equal sign.
The transitive property states that if A = B and B = C, then A = C. This ranks up there with reflexivity in how often it's used in geometry. To remember it, just build a little train that looks like A = B = C and conclude that the engine equals the caboose.
Okay, so that doesn't happen in real life, but it's a good way to remember the transitive property. In fact, you can build even longer trains like A = B = C = D, and conclude that A = D. Don't believe us? We can prove it (with a proof, no less).
Claim: If A = B and B = C and C = D, then A = D.
Proof: If we know A = B and B = C, we can conclude by the transitive property that A = C. If we also know C = D, then we have both A = C and C = D. One more use of the transitive property will finally give us A = D.
There's also the substitution property of equality. It says that if you know two things are equal, you can substitute one for another. Simple enough, right?
We used this property a lot in algebra. Sometimes we would solve for x, and then go back and substitute that number for x to figure out y. Remember that? The substitution property deserves a big thank you card.
Here's a handy list. (We know all these properties have ridiculously technical sounding names, but it's what they're called and we're stuck with it.)
- Reflexive Property: A = A.
- Symmetric Property: if A = B, then B = A.
- Transitive Property: if A = B and B = C, then A = C.
- Substitution Property: if A = B and p(A) is true, then p(B) is true. Here, p(A) is just any statement that has A in it, and p(B) is what you get when you replace A with B.