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46. Determine the truth value of the statement IxVy(r < y2) if the domain for the variables consists of 9) the positive real numbers b) the integers. C) the nonz...

Question

46. Determine the truth value of the statement IxVy(r < y2) if the domain for the variables consists of 9) the positive real numbers b) the integers. C) the nonzero rcal numbers_

46. Determine the truth value of the statement IxVy(r < y2) if the domain for the variables consists of 9) the positive real numbers b) the integers. C) the nonzero rcal numbers_



Answers

Determine the truth value of the statement $\exists x \forall y\left(x \leq y^{2}\right)$ if the domain for the variables consists of
a) the positive real numbers.
b) the integers.
c) the nonzero real numbers.

It's clear. So when you read here, so we have this, which we are given It's Y equals one. And this means that for every X in the domain, there's a value. Why in the domain such that Why, if the interests of X or there when you multiply them together, their product is one. So if we look at a the statement's true. So when the domain has all non zero real numbers, because the inverse of a non non zero real number X is why equals one over X and why is also, um, a non zero rial number? So which is equal to one? So, you know, the statement's true for part B, the statements falls. So when the domain has non zero integers, Um, since the inverse of a non zero into your ex is gonna be one over X, which is why then why is not a non zero and teacher for part C? The statement's true. So in the domain has all positive real numbers since the inverse of a positive real number is, um, real Number X is why, which is equal to one over X, then why is also a positive rial number

Well that's probably wanted to turn the truth value of each of the statement Now. The domain is all injured. What's on a They have for all in And swear it is greater than April 2. 0. Well, when you swear manager that is always greater than we oppose zero. So that is true on the we have there exists and in such that n squared is two but there is no integer who squares to. And so this is false. one C. We have for all in in squared is greater than or equal to him. This is not true for all real numbers, but this is true for all integers. The square of an integer is always bigger than the integer itself. And then for D we have their existing in Such that inspired is less than zero but there is no integer whose square is a negative number. And that means this is false.

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.

Yeah. That's probably going to determine the truth value of each state over the domain of all energy and so on. A. We have for all in and plus one is greater than it. And this is true because if you add one any integer, it is greater than what the manager was originally on B. We have their existing in Shots at two in three. We only need to find one to show that this is true because there exists and in a zero shows that that is true. And so it's a true state on C. We have their existent in such the end as negative end. And once again this is true and once again and a zero is our example that shows that it's true. And then lastly on the we have for all in three in is less than or equal to four in. Mm. And this is false. As a counter example. Take a negative number like end is -1. You put in -1. That will not be true. That's what since it's not true for that in it's not true for all him. And so that's false.


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