# Is the Statement True or False?: I Gave a Reason Why it is True

• Miike012
In summary, the conversation discusses the correctness of a statement regarding the existence of a multiplicative inverse for a real number. The statement is shown to be true when the universal quantification of y comes before the existential quantification of x, and false when the order is reversed. The value of x is not fixed, as it can vary for different values of y.

#### Miike012

I don't understant how the statement is false. Because I gave a reason why it is true. Can someone explain please? thank u

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Miike012 said:
I don't understant how the statement is false. Because I gave a reason why it is true. Can someone explain please? thank u

As long as y ≠ 0, it has a multiplicative inverse 1/y. Then if x = 1/y, xy = (1/y)(y) = 1.

Mark44 said:

As long as y ≠ 0, it has a multiplicative inverse 1/y. Then if x = 1/y, xy = (1/y)(y) = 1.

Yes, F is the "correct" answer
But I believe the correct answer is T

Miike012 said:
Yes, F is the "correct" answer
But I believe the correct answer is T

Well, it's not true. Pay attention the quantifiers, it says there exists a fixed x such that for all y. x can't depend on y.

Dick said:
Well, it's not true. Pay attention the quantifiers, it says there exists a fixed x such that for all y. x can't depend on y.

So if an existential quantification of x is before the univeral quantification of y then that means that the value of x must be fixed?

what if it was the universal quant of y then the existential quan of x? Then would x still have to be fixed?

The way I am reading it is...
There exists a real number x such that for all real numbers y not equal to zero , the expression xy = 1.

Or basically how I am interpreting it is, Let y = a where a is a real number and not equal to zero, then we can find a value of x such that xa = 1. The value of x that we are looking for is x = 1/a. Then choose a number y = b such that b =/= a and not equal to zero, then we can find a value of x such that xb = 1.
We would repeat this process for all values y = a where a is all real numbers.
So as you can see in my understanding of the sentence, the value of x is not "fixed" as u say it is.

##\exists x \forall y (y \neq 0 \implies xy = 1)##
is false.

##\forall y \exists x (y \neq 0 \implies xy = 1)##
is true.

## 1. Is it necessary to give a reason when determining if a statement is true or false?

Yes, providing a reason helps to support your claim and make your argument more credible.

## 2. How can I determine if a statement is true or false?

One way to determine if a statement is true or false is to gather evidence and evaluate it critically. You can also consult reliable sources and experts in the field.

## 3. What makes a reason convincing when determining if a statement is true or false?

A convincing reason is one that is logical, supported by evidence, and free from bias. It should also address any counterarguments and be relevant to the statement in question.

## 4. Can a statement be both true and false at the same time?

No, a statement can only be either true or false. However, there may be different interpretations or perspectives that can lead to conflicting opinions on the statement.

## 5. Why is it important to determine if a statement is true or false?

It is important to determine the truth or falsity of a statement in order to make informed decisions and avoid spreading misinformation. It also helps to promote critical thinking and intellectual honesty.