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Prove that the number of vertices in a full binary tree is odd....

Question

Prove that the number of vertices in a full binary tree is odd.

Prove that the number of vertices in a full binary tree is odd.



Answers

Prove that the number of vertices in a full binary tree is odd.

Still in this questionnaire is supposed to show that the product of an odd and even function is odd. The first step will be today ffx bee and our function and Dior Becks be eating, and you can choose any letter you want for the functions. That doesn't matter as long as there, too, functions with one order another. Even so, then, by the definition, often odd function. We must have f off negative eggs to be equal to ever act and g of negative X should give us your bag. Since, geez, even so now we define you have X to be the product of the ardent, even function toe here, Vaksince equal to ever time's Tio dio eggs. And that's a product still to find out whether the products of the order and he even function. It's odd we must shake. It satisfies the definition. Often our function, which is feels negative. X equals negative field legs. So in this next step we check what p of negative X so replacing the X values with the negative eggs. We have negative effort eggs because that's a definition for F over here and then we have a fix. But As you can see and affects NGO times, you have eggs. It's what we define earlier to be a product of lord and in function, so we can replace that with Yeah, thanks. So, looking closely, you can see that this is he of negative eggs, given the negative view of it. And that is a definition often odd function. So therefore we conclude that in product is art.

So we're even the information that and from the minus thanks. Goad you. Thanks and g of minus Thanks in cardiology. Thanks. So from here, every trying to end them off. So we should get the f thanks, plus g thanks. And now they want. Then we defend Surely is new friendship much banks on energy test If the agent is even a all 200. Yeah, Much off minus. Thanks. Now they could, you know, effort minus thanks. Plus, Jane off minus Thanks. I never go Jew by just why didn't you? Information here was again the half of it. Thanks, Jim. Thanks. And you go to the next service Amused And, uh, some, uh, even function a even something here in the other case here we have the half minutes. Thanks. You can go to minus X and G of minus X equals your ministry ex even that they have both. Okay. And now where do you find on duh No, no x really go, Judah, If I bless G X and now a minus like you could use the air four minus pressed g off minus angst. So you go to my house. F has g X because they're on. Hold on now You could, Your Highness in front if thanks plus g X. So you go to minus X so I shouldn't even be out. So this sum off Oh, function is aunt.

85. We know that f is odd, so we know that of negative X equals negative f of X because that's the definition of a non function. So if I put in f er negative zero that's gonna eat well naked in yeah of zero, which is the same thing is saying f zero e holes negative f of zero, which is the only number that the opposite of itself. So then we know that all negative functions that's through orjan zero.

Okay, this is problem number 69 and it's a proof we're asked to prove that JIA Becks is even if and only if f of X is even. And also G of X is odd if and only if f of X is odd. Where g of X equals one over f of x. Another way to think of that is G of X and F of X are reciprocal functions. So to do this proof thoroughly, incorrectly, it's really quite a long process. I'm not sure if your teacher or instructor is expecting you to do a purely theoretical mathematical proof, but I'm going to do it justice here and work through a pure mathematicians proof. So in order to prove an if and only if statement, which is called a by conditional statement, which means it's going in both directions, we need to prove two things. So I've broken it into part one. In part two. We need to prove that if f of X is even then, G of X is even, and we also need to prove that if G of X is even then f vexes even and if we can prove both of those things. Then we can conclude that G of X is even if and only if f of X is even. Okay. So here we go through part one of the first of two proofs. So buckle up. This is going to be quite a ride. So if f of X is even then g of X is even, so are given is that f of X is even. What does it mean for a function to be even? Well, it means that opposite X values will have the same. Why value? So we can say then f of X equals f of the opposite of X. That's a way to algebraic Lee State that found a function is even now we know that G of X is one over f of X. And according to the statement above, if F of X is equal to f of the opposite of X, I could do a substitution and show that G of X is equal to one over f of the opposite of X. Well, we also know that G of the opposite of acts just by definition is one over f of the opposite of X. If you just take the function G of X equals one over f of X and put the opposite of X in both places. Okay, well, look at what we have here. We have that won over F of the opposite of X is equal to G of X and we also have that one over f of the opposite of X is equal to G of the opposite of X. So we can substitute and say that G of X is equal to G of the opposite of X. And that tells us that g of X is even so. We've just proven that if f of X is even then g of X is even so. That part is done. Now we go the other direction part to prove if G of X is even than f of X is even so. This time are given is that g of X is even What does that mean and even function just like before? Opposite X coordinates have equal Why coordinates So we can say then g of X equals g of the opposite of X. Now what is G of X g of X is one over f of X. Well, If G of X is equal to G of the opposite of X, then one over F of X is also equal to G of the opposite of X. But what is G of the opposite of X? By definition, we know that G of the opposite of X is one over f of the opposite of X. Just by taking the relationship we had in the beginning and plugging the opposite of X in to both inputs. Okay, once again, we can do a substitution. We see that G of the opposite of X is equal to won over f of X, and it's also equal to one over f of the opposite of X. So that tells us that one over f of X is equal to one over f of the opposite of X. Well, from that we can conclude that f of X equals half of the opposite of X. And that means that f of X is even so. We've just proven that if g of X is even then, f of X is even. And in part one, we proved if f of X is even then g of X is even put them together and we have the complete proof that G of X is even if and only if f of X is even cool. Okay, that takes care of half of our problem. The other half is the same thing, but for odd. So here we go helpfully, As you see, the different parts emerge and you start to see the similarities. You'll begin to it'll it'll begin to kind of sync in, and you can see the logic behind what we're doing. So Part one is to prove if F of X is odd, then g of X is odd. So are given is that f of X is odd. Well, what does that mean for any odd function? Opposite X values have opposite. Why values? So that means that the opposite of F of X is equal to f of the opposite of X opposite X values have opposite. Why values. Now we know by definition that G of the opposite of X is one over f of the opposite of X. That's just from our original relationship we have at the top of the screen, but plugging in the opposite of X in both inputs, so I can replace F of the opposite of X with with the opposite of F of X and we have equals one over the opposite of F of X. Well, one over the opposite of F of X is the opposite of G of X. So if you follow this string of equivalent statements, we have that G of the opposite of X is equal to the opposite of G of X and that tells us that G of X is odd. Okay, so that half of the proof is done and we'll do something similar to prove that if G of X is odd, then f of X is odd. So are given is that g of X is odd. And what does that mean? Means that for G of X, opposite inputs have opposite outputs, so we can say the opposite of G of X is equal to G of the opposite of X. Well, what is G of the opposite of X g of the opposite of X is one over f of the opposite of X, so that is equivalent to the opposite of G of X, because if the opposite of G of X equals G of the opposite of X, then we can just substitute that one in there. Okay, well, what is the opposite of G of X? The opposite of G of X is one over the opposite of F of X. So now we have the opposite of G of X equivalent to a couple of different things. The opposite of G of X is equivalent to f of the opposite of X, and the opposite of G of X is equivalent to the opposite of F of X opposite of one over f of X. So we can set those equal to each other. One over f of the opposite of X is equal to one over negative opposite of X or opposite one over. That's I'm getting tongue tied one over the opposite of F of X is what I mean to say so from that we can conclude that f of the opposite of X equals the opposite of F of X. And that means that f is odd. Okay, so we have proven that if f of X is odd, then g of X Assad and if g of x Assad, then f of X is odd putting those together we have G of X is odd if and only if f of X is odd. And that completes problem number 69. Who if you stuck with me the whole time Congratulations. That was a lot.


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