Well, take a shot at these questions, and we'll give you feedback. Surely you suspect that the statement that starts with "For every two real numbers x and y..." is written as ∀x ∀y...
Hi guys,
I'm struggling to take on these questions where I am asked to write propositions using connectives and quantifiers:
Let P(x) be the statement that says that a real number x has some property P.
For every two real numbers x and y with x<y, there is a real number with the property P between x and y.
I must also construct a negation for this problem. There is also a very similar question to this that I have to do but instead of using P(x) as the statement that says a real number x has some property P, I must let P(n) be the statement that says that a natural number n has some property P then write the following statement using connectives and quantifiers:
Any sum m + n of natural numbers m and n which have the property P, has the property P.
Good start. You are missing something after the existence symbol. Right now, that reads: "For all real numbers x, for all real numbers y, x<y, there exists an inequality such that x<P(z)<y for some undefined z."
Try using a colon when you are restricting a quantifier. For example:
I would encourage the OP to expand the abbreviation . This is not the basic formula syntax, and it is important to be able to write it in full.
Often people are not sure whether "For all x and y such that x < y, Q(x, y) holds" is rendered ∀x ∀y. x < y ∧ Q(x, y) or ∀x ∀y. x < y → Q(x, y). The first version is wrong because it is not claimed that x < y for all x and y (and also Q(x, y)). Another way to look at it, if someone chose x and y and it happened that x ≥ y, then nothing is claimed; Q(x, y) is only guaranteed when x < y. This resembles implication because implication is true when the hypothesis is false. Indeed, the correct formula is ∀x ∀y. x < y → Q(x, y).
Do you mean the negation? Or the similar question with natural numbers? For the similar question with natural numbers, here is a start...
(I updated this to take emakarov's advice into account. I have seen and used a colon to restrict qualifiers, so I was not aware it was not "basic". Then again, I have never actually checked to see what is considered basic syntax...)
You change all "for all" signs to "there exists" signs, all "there exists" signs to "for all" signs, and negate any expressions based on those. So, let's negate emakarov's example:
It's negation would be: . So, how do you negate a conditional statement? The only time a conditional is false is when you have . So, it would be
Q is some property. Given x and y, Q(x,y) is true if the property holds for the given x and y and it is false if the property does not hold. In other words, replace Q(x,y) with another expression that completes the problem. If x<y, then what should be true? Q(x,y) = "there is a real number between x and y with the property P"
Looks good to me