Hello everyone,
I'm having problems with the following type of tutorial questions (yes, there are more of them):
Use logic notation to express the following:
"There is at most one x with P(x)"
I understand that to mean, "if there exists an x such that P(x), then for all other x not P(x)."
but this is wrong, or at least badly expressed. Would the following be right?
Apcalculus gave the right hint.
Translating sentences of the form "If there exists such that , then " is a little counterintuitive. One could try
but this would not be right because the scope of does not extend to . Another attempt is
This is also wrong because this proposition affirms the existence of something, while the original plain language sentence did not claim this.
The right way to translate "If there exists such that , then " is using the universal quantifier:
Well, look: you need to translate the following statement.
Instead of this, I'll translate a not-very-meaningful statement just to give an example:If there two x-values x1 and x2 such that P(x1) and P(x2), then x1 = x2
If there is an such that and , then .
This would be
Of course, to really solve the problem, besides writing the actual formula, you need to understand very well why
is equivalent to your original statementIf there two x-values x1 and x2 such that P(x1) and P(x2), then x1 = x2.There is at most one x with P(x)
No, I am referring to apcalculus's post.
I don't fully agree with your version.
The right expression of the original statement ("There is at most one...")? This is not correct because the original statement does not assert the existence of anything while your sentence does.
This is also strange because now instead of "There exist an x with P(x)" you write .This in logic in quantifier form is expressed as follows:
Yes, I understand the reason. That makes sense, but the quantifiers and the logical operators do not make sense. The problem is the overloading of the x-variable. That hurts my head. It's the formula-writing which is the problem, not the problem itself.
I have to admit, I find this very hard. Thank you all for your help, nevertheless.