Understanding Quantifier Statements: Homework Help and Solutions

[XodoX]

Homework Statement

http://img641.imageshack.us/i/mathy.png/

Homework Equations

The Attempt at a Solution

a) For every x(Q)x and for every xP(x) AND for every xQ(x) and for every xP(x).

-> Therefore, it is equivalent, right? Do I do it like this for the other ones, too?

b) Looks the same to me, so it's also equivalent ?

c) I think this statement is contradictory -> not equivalent.

 

[Robert1986]

Homework Statement

http://img641.imageshack.us/i/mathy.png/

Homework Equations

The Attempt at a Solution

a) For every x(Q)x and for every xP(x) AND for every xQ(x) and for every xP(x).

-> Therefore, it is equivalent, right? Do I do it like this for the other ones, too?

b) Looks the same to me, so it's also equivalent ?

c) I think this statement is contradictory -> not equivalent.

For the LHS of b to be true, we look at each x. if EITHER Q(x) is true or P(x) is true, the (Q(x) V P(x)) is true for that x. If (Q(x) V P(x)) is true for each x, then the LHS is true. OK, so, knowing that, what do you think the RHS of b means? Does it still look the same?

 

[XodoX]

For the LHS of b to be true, we look at each x. if EITHER Q(x) is true or P(x) is true, the (Q(x) V P(x)) is true for that x. If (Q(x) V P(x)) is true for each x, then the LHS is true. OK, so, knowing that, what do you think the RHS of b means? Does it still look the same?

Could you please clarify what LHS and RHS means? And, which one are you talking about?

 

[Robert1986]

Oh, sorry, I was talking about b, and LHS means Left Hand Side and RHS means Right Hand Side.

 

[XodoX]

Oh, sorry, I was talking about b, and LHS means Left Hand Side and RHS means Right Hand Side.

You're right, I misread the signs. It's or, not and. I still think it's equivalent.

For every xQ(x) or for evey xP(x) AND For every x(Q)x or for every xP(x). It's basically the same, to me. So, if the left side is true, the right side would be true,too, since it's the same statement.

 

[XodoX]

Oh, you mean, because it is or, it could be true or not. If one side isn't true, the other one could still be, and that makes it not equivalent?

 

[Robert1986]

Oh, you mean, because it is or, it could be true or not. If one side isn't true, the other one could still be, and that makes it not equivalent?

Yes, exactly. For example, let our set X be the set {1,2,3,4...}. Let for x in X, let Q(x) be true if x is even and let P(x) be true if x is odd.

Now, let's consider b:

For All x, (Q(x) V P(x)). This is certainly a true statement since for each x in X either Q(x) is true or P(x) is true since every element of X is either even or odd.


However, the statement For All x Q(x) OR For All x P(x) is not true. For this statement to be true, then every single x in X would have to be even or every single x in X would have to be odd. Clearly this is not true.

 

[Apphysicist]

(a) can also be clearly proven:

(1) Given: for all x, Q(x) AND P(x)
(2) Q(x) AND P(x) [because x is free for x, you can rip off the quantifier]
(3) Q(x) [follows from 2]
(4) P(x) [follows from 2]
(5) for all x, Q(x) [generalization]
(6) for all x, P(x) [generalization]
(7) for all x Q(x) AND for all x P(x) [follows from 5 and 6]


After that, coming up with a single model that raises doubt could be enough for you to understand...for (b):

Let Q(x) mean ON and P(x) mean OFF. Let all x be the universe of electronics.
LHS: any electronic device is either on OR off.
RHS: all electronic devices are on OR all electronic devices are off.

Are those the same statement?

 

[XodoX]

Thanks you two. Yes, I understand it now. Now that it has been illustrated. Was I right with c, though ( not equivalent) ?

 

[Apphysicist]

Thanks you two. Yes, I understand it now. Now that it has been illustrated. Was I right with c, though ( not equivalent) ?

Q(x):= x is big
P(x):= x is tall
Universe:= trees

For every x, if it is big, then it is tall. [If a tree is big, then it is tall.]

Versus

If every tree is big, then every tree is tall.

There's a model for you; I think it shows well.