HOw do u decode this statement to find its equivalent?

[mr_coffee]

Hello everyone, another pratice exam question I'm having issues on.

Which statement is equivalent to the following statement?
It is not true in general that two real numbers must be equal if their squares are equal.

Would i write an existence statement like

There exists 2 real numbers, x and y, such that if x = y then x^2=y^2.

But the statement is It is NOT true...so now would i take the negation of that existence statement and get...

~(There exists 2 real numbers, x and y, such that if x = y then x^2=y^2.) =
There are 2 real numbers, x, and y, such that x = y and x^2 != y^2.

But this isn't right, because the answer is the following:

THere are two different real numbers that have the same square.

I don't see how they came up with this. Any help would be great.

 

[HallsofIvy]

It is not true in general that two real numbers must be equal if their squares are equal.

Decode? This isn't a code, it is a statement. You don't rephrase it by following some formula, you do it by thinking! Notice that the last part of it is "if two squares are equal". It is not true that "if x²= y² then x= y". Note that the "if" part is not your "if x= y" but "if x²= y[/sup]2[/sup]".

If you must apply formulas without thinking, then use this: the negation of "for all x, ..." is "for somex, not ...".
The negation of "for all x, y if x²= y[/sup]2[/sup] then x= y" is "There exist x, y such that x is not equal to y but x²= y[/sup]2[/sup]."

 

[mr_coffee]

Thanks for the clarification.

It is not true in general that two real numbers must be equal if their squares are equal.

I thought this was an existence statement, not a universal.

but is it universal? its not saying for all real numbers, but for just 2, that's why i thought they ment existence rather than universal.

if its universal would it be this?
For all real numbers x and y, if x^2= y^2 then x = y.


But I'm not sure where I would be the "not true" part, or does the "not true" mean to take the negation of the statement?
so it would be

There exists real numbers x and y, such that x^2 = y^2 and x != y.

I believe this negation is correct if it was a universal statement but it can't be right because its not the answer on the exam. So either the "it is not true" part is messing me up or I used a universal statement where I should be using an existence statement. But I'm not sure which.


Or does There exists real numbers x and y, such that x^2 = y^2 and x != y. Logically equivlent to saying: There are two different real numbers that have the same squares? Since this is existence, and not universal i see where they are getting "there are two different real numbers" and its not saying x == y, but saying they are not equal, but it is saying x^2 = y^2 so maybe it is right.

 

[HallsofIvy]

Thanks for the clarification.

It is not true in general that two real numbers must be equal if their squares are equal.

I thought this was an existence statement, not a universal.

but is it universal? its not saying for all real numbers, but for just 2, that's why i thought they ment existence rather than universal.

It says "in general". Yes, it talks about two numbers but they can be any two numbers. That's why this is a universal statement.

if its universal would it be this?
For all real numbers x and y, if x^2= y^2 then x = y.

Yes, that's exactly what it says.

But I'm not sure where I would be the "not true" part, or does the "not true" mean to take the negation of the statement?

Yes, the statement is "It is not true in general that ..." and then the entire statement. It is the negation of the entire statement

so it would be

There exists real numbers x and y, such that x^2 = y^2 and x != y.

I believe this negation is correct if it was a universal statement but it can't be right because its not the answer on the exam.

Tnen what was the answer on the exam?

So either the "it is not true" part is messing me up or I used a universal statement where I should be using an existence statement. But I'm not sure which.


Or does There exists real numbers x and y, such that x^2 = y^2 and x != y. Logically equivlent to saying: There are two different real numbers that have the same squares? Since this is existence, and not universal i see where they are getting "there are two different real numbers" and its not saying x == y, but saying they are not equal, but it is saying x^2 = y^2 so maybe it is right.

That last sentence completely confuses me! If you are saying that the answer on the exam was "There are two different real numbers that have the same squares", the is exactly the same as your "There exists real numbers x and y, such that x^2 = y^2 and x != y".

I would be inclined to say "There are two different real numbers that have the same squares" is a better answer than "there exist real numbers x and y, such that x^2= y^2 and x != y" because it is worded in the same way as the original statement "It is not true in general that two real numbers must be equal if their squares are equal" rather than "It is not true, for all real numbers x, y, that if x^2= y^2, then x= y".

 

[mr_coffee]

Thanks Ivey!

Yes the answer was:
There are two different real numbers that have the same squares.

Sorry I ramble a lot in my posts, I shall try to minimize that.