I'm working on a set of True and False statement that deal with quantifiers, and I suspect there may be an error. The universal set here is rational numbers, so almost everything is on the table. I've triple checked each one with what I could find online, but apparently at least one of these is not the correct answer. Here is my reasoning for each:
- True as y = 1 - x
- False as there is no singular value such that when multiplied by any other real number, their product is 1
- False as it violate the communicative property of addition, which is true for all real numbers
- False as there is no spot the two equations meet, meaning both equations can never be true at the same time
- True as for all rational possibilities of $x^2$, there will exist a rational square root.
- Impossible at there is no rational number that can be squared to make a negative number
- The statement says that for every two real numbers, there exists a real number that is the average of the two real numbers. This is always true and thus the statement is true.
- True as for all rational values of x, there exists a singular value of y = 0 to make a product of 0
- False as there is not a rational value of y that solves the equation for all values of x.
- False since for negative rational numbers, there is no value of y that could be squared to create it.
- True as the square root of 2 is a real number.
- True as for all values of x, there will exist a y such that xy = 1. This is shown with the inverse property of multiplication.
Any hints or help would be greatly appeciated.
Nr. $7$ is false. It does not say for each pair of real numbers there is a real number that is their average; that would be $\forall x\forall y\exists z\,\left(z=\frac{x+y}2\right)$. It says that for each real number $x$ there is a real number $z$ such that $z=\frac{x+y}2$ for every real number $y$, which is clearly false: that says that for each real number $x$ the function
$$f_x:\Bbb R\to\Bbb R:y\mapsto\frac{x+y}2$$
is a constant function.