Study Guide

Logic and Proof - The Quality of Equality

The Quality of Equality


Understanding equality, or sameness, is a universal theme in all areas of mathematics. When mathematicians say, "2 + 2 = 4," they mean that the two things on either side of the equal sign are literally the same thing: they are two ways of describing the same object. They're equal.

In mathematics, the objects we work with are usually numbers. The thing is, we can't usually see them (talk about imaginary numbers, huh?), and we're forced to make do with looking at their descriptions instead.

Somebody with too much free time might choose to write "(1 + 1 + 1 + 1) – (1 + 1 + 1) + 1 + (1 – 1 – 1) – (1 – 1) + (1 + 1)" instead of the significantly easier to read "3," even though they describe the exact same number. Way to mess with us.

  • Properties of Equality

    So how do we really know when two things are equal? We rely primarily on 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're 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).

    Sample Problem

    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.
  • Arithmetic Properties

    When talking about numbers (which we usually are when we use the equal sign), we have a few extra properties of equality that come from arithmetic.

    The addition property of equality states that if you have numbers A, B, and C such that A = B, then you also know A + C = B + C. That is, you can add the same number to both sides of the equation. But only if you want to.

    Sample Problem

    Let's say that A = B. Which properties are used in making the statement B + C = A + C?

    Well, if we start with A = B, the first thing we did is switch their positions around and write B = A. That's the symmetric property's doing. Then, we added C to both sides. We can do that according to the addition property. So in total, we used the symmetric and addition properties.

    Similarly, the subtraction property says that if A = B then AC = BC, and the multiplication property says that if A = B then AC = BC. Finally, the division property says that if A = B and C is nonzero, then AC = BC. Basically, as long as you do the same exact thing to both sides of the equation, you should be fine.

    But whatever you do, never divide by zero. You may already know this, but dividing by zero is very bad. Many esteemed mathematicians and physicists believe that it will create a black hole in the center of the earth that'll swallow the whole universe and erase everything in existence before you can even reach for your eraser. Avoid it at all costs.

    Sample Problem

    If A = B and B = 4, and C = D and D = 0, can we say that AD = 4C?

    At first glance, everything looks hunky-dory. Applying the transitive property allows us to say that A = 4, and since we're given that C = D and the division property allows us to divide both sides by the same number, where's the problem?

    Wait…does that say D = 0? Uh oh. Abandon ship! Abort the mission! Reject all statements of the sort because dividing by zero would be a huge, disastrous mistake.

    Some people lump the distributive property in with properties of equality, but really it's more a property of addition and subtraction. In any case, we might as well tell you what it is now that we're talking about it.

    The distributive property says that if A, B, and C are numbers, then A(B + C) = AB + AC. In other words, you distribute the A to all the things inside the parentheses. Fair is fair.

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