Deductive and inductive reasoning
Inductive reasoning (example 2)
Luis looked at the expression x minus y times x squared plus xy plus y squared, and wrote the following. He said, using the distributive property-- so he took x minus y and he distributed this expression onto the x and this expression onto the negative y. So that's why we have an x times this entire expression over here. x times the entire expression minus y times this entire expression. Then he says, using the distributive property again, this is equal to-- so then he distributed this x into that and got all of the first three terms here. He got these first three terms. And then he distributed the y. He distributed the y over here and he got these three terms. And then he saw, looks like that this term x squared y cancels out with a minus x squared y, and the xy squared cancels out with the negative xy squared. Then he saw that it equals x to the third minus y to the third. And he wrote down, I conjecture that x minus y times x squared plus xy plus y squared is equal to x to the third minus y to the third for all x and y. Did Carlos use inductive reasoning? Explain. Well, inductive reasoning is looking at a sample of things-- looking at some set of data, if you will-- and then making a generalization based on that. You're not 100% sure, but based on what you've seen so far, you think that the pattern would continue. Or you think it might be true for all things that have that type of property or whatever. Now in this situation, he didn't look at some type of a sample. He actually just did a proof. He multiplied this out algebraically. In fact, it's incorrect for him to say I conjecture here. A conjecture is a statement or proposition that is unproven, but it's probably going to be true. It's unproven but it seems reasonable, or it seems likely that it's true. This isn't a conjecture. This is proven. He proved that x minus y times x squared plus xy plus y squared is equal to x to the third minus y third. He should have written-- and this is a much stronger thing to say-- he should have said, I proved that this is true for all x and y. So to answer the question, did he use inductive reasoning? No. I would say that he made an outright proof. No, he made a proof. Inductive reasoning would have been, if he would have saw, if you would have given him 5 minus 2 is equal to-- or 5 minus 2 times 5 squared plus 5 times 2 plus 2 squared, and you saw that that was equal to the same thing as 5 to the third minus 2 to the third. And then let's say he did it for, I don't know, one in seven in a couple of examples. And it kept holding for all the examples that it was the first number cubed minus the second number cubed. Then it would have been inductive reasoning to say that that is true for all numbers x and y. But here it's not an induction. He didn't use induction. Or I shouldn't say induction. He didn't use inductive reasoning. He outright proved that this statement is true for all x and y.