Proving

[LukeeVassallo]

Hi, I have this proof and I am mainly confused by the words "there exists / for some" (there is the inverted E symbol used to denote that). This is the statement: " For all real values of y, there exists a real value x, such that x^2 = y" .... The question states that I need to say if this is true and write it's negation. The negation I have no problem, but for me to be sure it's true I need to do a convincing proof. Any tips ? Many Thanks - Luke

 

[pka]

Hi, I have this proof and I am mainly confused by the words "there exists / for some" (there is the inverted E symbol used to denote that). This is the statement: " For all real values of y, there exists a real value x, such that x^2 = y" .... The question states that I need to say if this is true and write it's negation. The negation I have no problem, but for me to be sure it's true I need to do a convincing proof. Any tips ?

Well, it is certainly not true in the real numbers.
Consider \(\displaystyle y=-1\), what \(\displaystyle x\) has the property that \(\displaystyle x^2=y~?\)

 

[JeffM]

Hi, I have this proof and I am mainly confused by the words "there exists / for some" (there is the inverted E symbol used to denote that). This is the statement: " For all real values of y, there exists a real value x, such that x^2 = y" .... The question states that I need to say if this is true and write it's negation. The negation I have no problem, but for me to be sure it's true I need to do a convincing proof. Any tips ? Many Thanks - Luke

What is the problem exactly? Is it something like, write the negation of P and determine whether P or the negation of P is true, where P is the proposition given in your question?

What you have said makes no sense. You say you have a proof of P, but that is impossible because P, as you written it, is not true.

 

[HallsofIvy]

There is nothing in what you write, "say if this is true and write it's negation" that requires you to prove anything. As pka says, the statement "For all real values of y, there exists a real value x, such that x^2 = y" is NOT true. It's negation is "It is not true that for all real values of y, there exists a real value x, such that x^2 = y" or equivalently, "there exist a real value of y such that, for all real values of x, x^2 is NOT equal to y". THAT can be proven by exhibiting such a y, as pka did.