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

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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.
  • #1
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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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  • #2
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
Is F the "correct" answer?

As long as y ≠ 0, it has a multiplicative inverse 1/y. Then if x = 1/y, xy = (1/y)(y) = 1.
 
  • #3
Mark44 said:
Is F the "correct" answer?

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
 
  • #4
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.
 
  • #5
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?
 
  • #6
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.
 
  • #7
##\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.

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