In math, we can't explain that something is true just because. That's not enough. In this lesson, we'll learn to prove things using one of our most straightforward tools, the direct proof.
In the Mathematics Justice System, the truth is represented by two separate, yet equally important groups: the math police, who investigate possible crimes against mathematics, and the direct proofs that determine whether statements are true. These are their stories.
That's right; it's time to be math police. You can be the rookie straight from the academy. I'll be the veteran cop trying to teach you a thing or two, though maybe you end up teaching me in the end.
Anyway, in this lesson, we're going to use direct proofs to get to the bottom of some math mysteries. A direct proof is a method of showing whether a conditional statement is true or false using known facts and rules.
A conditional statement is an 'if, then' statement. We might say if p, then q, where p is our hypothesis, and q is our conclusion. We can show this like this:
This is how to show a conditional statement.
We know p is true, but we need to find out if q is true.
You see conditional statements all the time. If you stay up too late, then you'll be tired tomorrow. Or if you give a mouse a cookie, then he'll want a glass of milk.
When we're completing direct proofs, our statements don't even need to include the words 'if' and 'then.' Consider this one: The product of any two odd integers is odd. How is that a statement we can prove? We could rephrase it as:
If x and y are odd integers, then x * y results in an odd integer.
Maybe we know we have a murder victim. We need to prove whether or not the shady husband did it.
Sample Proof #1
Ok, but murder is outside of the math police's jurisdiction. With direct proofs, we're more likely to see something like this:
If a and b are both odd, then a + b is even.
So, our p is 'if a and b are both odd.' Our q is 'then a + b is even.' Our direct proof will be a series of statements that gets us from p to q. We need to fill in the gaps, like putting the pieces together when solving a murder.
Got your crime scene face on? Let's do this. We build our proof as a list, with each step on its own line.
First, we'll state, Suppose a and b are odd integers. That's the first half of our statement, or our p.
Next, we'll state, Then a = 2k + 1 and b = 2l + 1, where k and l are integers. Ok, rookie, are you with me? No? By definition, an even integer is just two times some integer. For example, 8 is just 2 * 4. And an odd integer is two times some integer, plus 1. Think of any odd integer, like 11. 11 is just 2 * 5, plus 1. So, we're picking variables for integers, k and l, and defining a and b in terms of them. If a is 11, then a is 2*5 + 1, right? Right.
Next, we can say, Therefore, a + b = (2k + 1) + (2l + 1) = 2(k + l + 1). We want to find out what a + b is - always keep your focus on the end goal. We just substituted here, then simplified.
Now we can say, If k and l are integers, so is k + l + 1. Why? We already know k and l are integers. What about 1? Yep, also an integer. So, the sum of three integers is also an integer.
Finally, we can say, a + b is even. That's what we wanted to prove. We just solved the case. Did you follow me? a + b must be even if it's equal to 2 times an integer. Remember earlier when we said that an even integer is just two times an integer? We just showed that a + b is equal to two times an integer, so this case is closed. Time for a doughnut and the closing credits:
Yep, like that.
Sample Proof #2
Well, enough down time. You know what comes right after an episode of Law & Proofs? Another episode of Law & Proofs. Let's solve another case. We're investigating a series of bank robberies and... wait, sorry, wrong show. How about this? If a and b are odd integers, then ab must also be an odd integer. Whew. That's more in our wheelhouse.
So, what's our hypothesis? 'If a and b are odd integers.' That's our p. Our conclusion, or q, is 'then ab must also be an odd integer.'
Wait, you might say. This one's easy. If a is 1 and b is 3, then ab is 3. That's odd. If a is 5 and b is 11, then ab is 55. Again, it's odd. Sure, that's all true. But do you want to do a proof where we show that this statement is true for every single set of odd numbers? How long is your shift?
We want to do a direct proof that simply proves the statement for all odd integers.
Let's start, naturally, at the beginning: If a and b are odd integers, then a = 2x + 1 and b = 2y + 1, where x and y are integers. Ok, that looks complicated, but let's substitute some numbers to see what we're doing. We're saying x and y are integers. If x is 4, then 2x + 1 is 9. If x is 7, then 2x + 1 is 15. No matter what x or y are, that 2 will make it even, then that '+ 1' will make it odd. And you thought '+ 1' was just for wedding invitations.
It's important to use two integers, like x and y, and not just x. Why? Because a and b are different integers.
Next, we can state, ab = (2x + 1)(2y + 1) because of the definition of ab. We're trying to prove that ab is odd, so let's take our ab values and multiply them together.
Next, we do some math. ab = 4xy + 2x + 2y + 1 by expanding the brackets. Then, we get ab = 2(2xy + x + y) + 1 because 2 is a common factor.
That's it. Do you see it? 2xy + x + y is going to be an integer. It doesn't matter how complicated it looks; the result will be some integer. So, we have 2 times an integer, then plus 1. That's how we defined an odd integer. So, ab must be odd. Another case closed.
In summary, we learned that police work involves a surprising amount of paperwork. More importantly, we learned about direct proofs.
A direct proof is a method of showing whether a conditional statement is true or false using known facts and rules. Conditional statements are 'if, then' statements. It's basically if p, then q. P is the hypothesis, and q is the conclusion. The direct proof is a series of statements that start with the hypothesis, then use known facts and processes to determine the truth of the conclusion.
You should have the ability to do the following after watching this video lesson:
- Define conditional statements
- Explain how to use direct proofs to show whether a conditional statement is true or false