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Determine the truth values of each of these statements if domain of each variables contain all real numbers VxVy (x/y= [) 2.VxJy (x/y= [) 3.3xVy (x y= 0)....

Question

Determine the truth values of each of these statements if domain of each variables contain all real numbers VxVy (x/y= [) 2.VxJy (x/y= [) 3.3xVy (x y= 0).

Determine the truth values of each of these statements if domain of each variables contain all real numbers VxVy (x/y= [) 2.VxJy (x/y= [) 3.3xVy (x y= 0).



Answers

Determine the truth value of the statement $\forall x \exists y(x y=1)$ if the domain for the variables consists of
a) the nonzero real numbers.
b) the nonzero integers.
c) the positive real numbers.

Thank you for all your existence. Wife. That explains why. Well, is the statement always true? Yeah, because, um ah, we can just find saying bye to they wouldn't have except is Or if X is one that are wise one Alexis too. Then our wise for solving equations based on where exes are wise, that value cleared part be. We have that. We're all experiencing this. Why, That's got X equal to y click. Okay, well, I see if you think about this. What about square root of negative millions? Well, it isn't possible for a negative number, because the real quick routes does not exist. This is the minute for see? We have this extra wise. It's the x Times. Why? But a statement it's drill could send. What if we'll explain all these times? Any number is always there. Do you have there and why it's that X Why did not equal to y. Well, we know that the property of uh addition is committed. So you should be able to hold to the statement is on people. Now we have a which is for all existence. Excellent example of that. This implies that there exists and why it's an X times. Why? Because yeah, basically whatever value we have for excessively just takes in verse of X and let that equals y. And then X times are inverse. Selects is equal to one. It's a joke. Or if we have this awesome X for all Why's that? Why's that? He could sculpt. This implies that X cover Y. Okay, so this statement is that there is a rule number six for every non sterile room number. Their product. It was one. But this is also not true. So the same minute now we have four all exists. And why such that ex wife? Good one. Yeah, well, we can set our why. To be equal to X minus one. Been a position of x explain. It's one well equal one, actually. No, I don't think you should be. Why? People too? Yeah, well, it Why equal to one when it then we have that expert 11 Chances are X, which is equal to one. So this exists for some valuable I such that that values why you want mine. You know, for eight you have their system extended some wine which that exports to one it was two and two extra clips for. Why? Because we only find one number work. Okay, but if we know it's like, uh two for For we noticed this portion is parallel with this portion, so they're obviously never gonna interact. It is still very that's the issue. Well, then I'll support I for all extra exists wife. That explains why you could, too. And two extra minutes. Why one? So if you draft, it's actually you get that's his graph into sex one. But in the statement, we fit for all exes should be working. So the statement is actually close, since it only into sex once at one comma, one now for day, we have for all for all. Why did you just see that he is able to extra twine over tip who were asking that the sea is the midpoint of X and white and it's too right if we have some really exes, and that's why we can draw a line under it. And then their existence means it's a expert Floyd over to is the midpoint

Gets arrested German troops Sally first. And then there's this wrong. Why such that exit Mr Meet with your wife. If the remains of the variables consists of a all of your own number, well, it's just a Jew. Are is positive, you know, another great well, but saying that there those are just some number except that statement is true. So we have that thanks equal to y when you're actually upset. Why equal to the square root of X over too? That's why I said to be positive. His ex is positive. Then if you square this, we get why Square two people two X over four. That's next. So this squares left in X and the ex can't be X is not the values for which the same in the chip. So this is a contradiction and we could do that. It is not just about you, except that the minister for Part B we have. There's just some exit that for all Why Wycliff? Well, we'll see Well if why's an engineer that every value or every institute square to steal an interview so we can find some value except says is true? It was on patrol. See, we have non zero real number. What happens if actually you don't We have that hero is estimated to Why squared? Actually, we don't have doors in our domain, but we confuse any value listings. So we can say that you get one is left twice square in there to make it to y squared. You know, if we have Gerald and that's it, there was Listen, any value, but a fine. This is actually true. Could you know that the square of our value is always positive and any value of X can be negative because square of this is gonna be positive.

Okay. The first sentence says there exists a real number X. Such that when you cubit you get -1, that's true X is -1. There exist a number here it is that you can cube and get -1. Alright. The next one says there exists a real number X. Such that when you raise it to the 4th power it's smaller than when you raise it to the second power. That's also true. Here's one X equals one half, one half to the fourth is 1/2 48 16 1/16 and one half squared is 1/4 and 1/16 is less than 1/4 Robert Just says there exists one here. One is There exists lots of them. Any fraction between zero and 1 would work there. But that's that's okay. It's still true because it says I can think of one. I did. Okay. The next one says for every X. Or for all X. When you square the opposite you get the same thing is when you square the number can remember X. Here is a real number. So this one is true also, okay minus X squared means negative X times negative X and the negative times a negative is a positive. Okay, so that's true for every X, no matter which one you tell me it's true. Okay, now you can't you can't prove it by giving examples because you have to say it works for every single one. So you have to be able to prove it in some kind of algebraic away like that. Okay, the next one says for every X when you multiply it by two, you get something bigger than it. That's false. And all I have to do is show you one. If I can show you one then the for all it's not true. How about X equals minus one. Two times -1 is not greater than -1. Okay, so for there exists, you have to think of one Prove not for all. You have to think of one to prove. For all, you have to do it algebraic lee. All right. I hope that helps. It's fine.

Well, this problem. Our domain is all real numbers and we want to determine the truth value. Each of the following statement. Now a we have there exists an X. Such that X squared is equal to two. So in other words, there is a real number such that X squared is to and that is true. There is a real number that if you square it gives you two namely ricin on B we have there exists an X. Such that X squared isn't a uniform and that is false. There is no real number. If you square it will give you a negative one. So that is not true. Let's see. We have for all acts X squared Plus two is greater than or equal to one. And that is true if you're taking your real number square it and and to that's larger than or equal to one. And so that history No, indeed we have for all that's X squared does not equal eps, that's not true. Okay, this is false. And all we need to do is present a counter example here Namely X is one the square X. And if you want to clearly acts as a real number, that's what here is our truth value these days


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