Lesson ten- Proofs in the Holistic Method

This lesson will explore different methods of proving, including how to get a proof started. If someone asks you to prove something, this lesson will show you how to get started.

Example: Someone asks you to prove 1 + 2 = 2 + 1.

What do you have to do, and how do you get started?

Step One: Check for axioms and give assumptions

Are there any given assumptions in the equation? No, not really, so check for axioms. Where could we find some axioms? What sort of field does this concern? Maybe it has something to do with algebra so, look for axioms in your algebra book. What do you find?

Step Two: Find an axiom where the operators and the sequence of variables match

As you know there are a number of axioms to choose from. Evaluate the following axioms to see if we should use them. Do they compare well to the equation: 1 + 2 = 2 + 1?

Communities axiom of multiplication: x • y = y • x, for any real number x and z

How well does this axiom compare to the original equation? Is the number of variable the same? Yes, good. Is the main operator the same? The order of the two equations matches, but the operation does not. So you should not choose this one for your axiom because the operator is wrong.

What about this axiom?

x • (y • z) = (z • y) • x, for any real number x, y, and z.

Would this be a good axiom to use for a proof? Are the number of variables the same? Is the operator the same? No. This axiom is worse than the last because the number and the operator are different.

What about this axiom?

x + y = y + x, for any real number x and y.

Would this be a good axiom to use? Are the number of variables the same? Yes. Is the main operator the same? Yes, so this would be a good one because the number and the operator match.

It is the communicative property, which states x + y = y + x.

Step Three: Use replacements to make the equation the same as the axiom

Use this law: x + y = y + x.

Replace x with 1

1 + y = y + 1

What this statement says is that for any x and y, you can replace x and y with any combination of numbers within a set. There is a wide variety possible, as you will see below. Why is this replacement possible?

If you have x + y = y + x, for all x and y, you can have 1 + 2 = 2 + 1, or 2 + 3 = 3 = 2, or 5 + 6 = 6 + 5. This is a large set of statements. So you can just pick one element of the large set. Remember that the set is a collection of equations. In other words, the axiom describes an infinite set of equations that are all true. When I say replace x + y = y + x, I mean just to replace that axiom with one of the endless equations. This in a way is set theory.

When you replace x with 1, you get the equation 1 + y = y + 1. In reality we used a second law to do this. X shows up one the left side of the equation, and x shows up on the right side. Is this actually the same x? Can x mean something else on the left side as it does on the right side?

As you know x does not mean the same thing everywhere. Just because we are using x in the equation x + y = y + x, does not mean that x means the same thing in the equation x • y = y • x. This is because x stands to represent a value and that value can change depending on equations. It may seem easy to believe that this x and another x are the same, but they may be different. So we reference the reflexive property on page 675.

The reflexive property states that within the same equation, a variable such as x will have the same value. In spoken language, Sarah = Sarah is not always true. You may be talking about more than one Sarah. One Sarah does not equal another Sarah. Or one iPad may have different apps than another iPad.

In algebra, however, it is safe to assume that within an equation, everything that has the same name is the same: axiom x = x

What do you think we need to do now to make 1 + 2 = 2 + 1 equal to 1 + y = y + 1?

Step Four: Replace y with 2

1 + 2 = 2 + 1

Now we have 1 + 2 = 2 + 1. Are we finished? Is it the same?

To review, we began with an axiom (x + y = y + x), which we can assume to be true. We then used replacements, which are okay to according to the reflexive property. According to set theory, it is okay to choose any range of values of x and y.

While the equation 1 + 2 = 2 + 1 looks valid and complete, is it the same? How can we make sure that this equation that we just arrived at is the same as the one given?

Step Five: Compare the result with the given expression letter by letter

If all characters match, they are the same.

To review

If you have to prove an argument, even outside the field of mathematics, check which field the problem falls within.

Next check for any axioms that apply to that problem. Find what is commonly assumed to be true, and make sure that it an acceptable axiom.

Next make sure the axiom fits your proof. If the number and order of operations are the same, then it’s okay. Then make replacements, because replacements are generally okay.

Note: There is a subtle difference between a and the.

a defines a new number. It introduces something that was not there before.
therefers to the previously defined number.

In algebra we use a and the to define and refer back to these values.

Example 1

A number c is added to 1 (this is a new number) c + 1
The number c has a value of 2 2 + 1
What is the value of the expression?

Example 2

The number c now has a value of 3. 3 + 1
What is the value of the expression?

Do you see how a and the have different meanings in word problems?

Example 3

A number c is added to 5 c + 5

What is the value of the expression?

You cannot be sure because using A in the previous sentence introduces a new value. That value has yet to be determined, and c is currently undefined. So when you read a, that usually means a new definition, which means forget about what you have read before.

Given the equations

x + y = y + x, if x and y are real numbers.

x • y = y • x, if x and y are real numbers.

Does each x above represent the same number even? No, because they are not in the same equation! Remember that variables within the same equation are the same, but variable among different equations can be different.

At this level of algebra, many of our problems contain symbols. In an upper-level math book, however, many of the problems contain words.

Therefore the laws may read:

The sum of a number, x and a number, y, equals the sum of the number, y, and the number, x if x and y are real numbers.

The product of a number, x and a number, y equals the product of the number, y and the number, x if x and y are real numbers.