quantifier calculus

[brainoverload]

Hi,

Im no math student or any related field for that matter. I made a mistake by registering for additional subject and despite being told that is about how computer works its just math and logic without relating it to computers in any way. To pass this i have a list of tasks but the most problem I have with it is this task. I have to prove that statetment is true or false. Here is the list:

(a) ∀x(p(x) ∨ q(x)) ⇒ ∀xp(x) ∨ ∀xq(x)
(b) ∃x(p(x) ∧ q(x)) ⇒ ∃xp(x) ∧ ∃xq(x)
(c) (∀xp(x) ⇒ ∀xq(x)) ⇒ ∀x(p(x) ⇒ q(x))
(d) ∃x(p(x) ⇒ q(x)) ⇔ (∀xp(x) ⇒ ∃xq(x))
e) ∀x∼∼(p(x)) ⇔ ∼ ∃x∼p(x)
(f) ∃xp(x) ∧ ∃x∼q(x) ⇒ ∀x(p(x) ∨ ∼q(x)

Big thanks in advance for help. I have to do it till monday, I know i posted this late but its my last resort.

 

[stapel]

Im no math student or any related field for that matter. I made a mistake by registering for additional subject and despite being told that is about how computer works its just math and logic without relating it to computers in any way.

Um... computers run on logic, which runs on math.

To pass this i have a list of tasks but the most problem I have with it is this task. I have to prove that statetment is true or false. Here is the list:

(a) ∀x(p(x) ∨ q(x)) ⇒ ∀xp(x) ∨ ∀xq(x)
(b) ∃x(p(x) ∧ q(x)) ⇒ ∃xp(x) ∧ ∃xq(x)
(c) (∀xp(x) ⇒ ∀xq(x)) ⇒ ∀x(p(x) ⇒ q(x))
(d) ∃x(p(x) ⇒ q(x)) ⇔ (∀xp(x) ⇒ ∃xq(x))
e) ∀x∼∼(p(x)) ⇔ ∼ ∃x∼p(x)
(f) ∃xp(x) ∧ ∃x∼q(x) ⇒ ∀x(p(x) ∨ ∼q(x)

Big thanks in advance for help. I have to do it till monday, I know i posted this late but its my last resort.

Okay, show us your thoughts and efforts so far, so we can see where you're having difficulties and can reply with assistance.

Please be complete. Thank you!

Eliz.

 

[pka]

Im no math student or any related field for that matter. I made a mistake by registering for additional subject and despite being told that is about how computer works its just math and logic without relating it to computers in any way. To pass this i have a list of tasks but the most problem I have with it is this task. I have to prove that statetment is true or false. Here is the list:
(a) ∀x(p(x) ∨ q(x)) ⇒ ∀xp(x) ∨ ∀xq(x)
(b) ∃x(p(x) ∧ q(x)) ⇒ ∃xp(x) ∧ ∃xq(x)
(c) (∀xp(x) ⇒ ∀xq(x)) ⇒ ∀x(p(x) ⇒ q(x))
(d) ∃x(p(x) ⇒ q(x)) ⇔ (∀xp(x) ⇒ ∃xq(x))
e) ∀x∼∼(p(x)) ⇔ ∼ ∃x∼p(x)
(f) ∃xp(x) ∧ ∃x∼q(x) ⇒ ∀x(p(x) ∨ ∼q(x)

The first difficulty that I see is the lack of a definite domain of discourse.
Are you told that or are you free to give your own?
For example, in a) suppose you have a bin of [imath]100[/imath] marbles, half red & rest blue.
[imath]\left( {\forall x} \right)\left[ {r(x) \vee b(x)} \right] \Rightarrow \left( {\forall x} \right)\left[ {r(x)} \right] \vee \left( {\forall x} \right)\left[ {b(x)} \right][/imath] Is that TRUE OR FALSE?

 

[Dr.Peterson]

Im no math student or any related field for that matter. I made a mistake by registering for additional subject and despite being told that is about how computer works its just math and logic without relating it to computers in any way. To pass this i have a list of tasks but the most problem I have with it is this task. I have to prove that statetment is true or false. Here is the list:

(a) ∀x(p(x) ∨ q(x)) ⇒ ∀xp(x) ∨ ∀xq(x)
(b) ∃x(p(x) ∧ q(x)) ⇒ ∃xp(x) ∧ ∃xq(x)
(c) (∀xp(x) ⇒ ∀xq(x)) ⇒ ∀x(p(x) ⇒ q(x))
(d) ∃x(p(x) ⇒ q(x)) ⇔ (∀xp(x) ⇒ ∃xq(x))
e) ∀x∼∼(p(x)) ⇔ ∼ ∃x∼p(x)
(f) ∃xp(x) ∧ ∃x∼q(x) ⇒ ∀x(p(x) ∨ ∼q(x)

Big thanks in advance for help. I have to do it till monday, I know i posted this late but its my last resort.

A proof depends on some set of axioms, theorems, rules, or something you already know to be true. There are somewhat different ways to present these, so we can't help you without knowing something of what you have been taught. (That's part of what we hope to see in the work we ask you to show.)

Please show us whatever you have been given to start with in writing proofs, and perhaps an example of the sort of proof you are expected to write.

Then show us your own thinking about at least one of these.

 

[brainoverload]

I dont think there is domain of discourse. Im attaching the example we solved at the lecture. T means that for example x>3 can be true or false and F means its false. In this example it was said that if p(x) is > 3 and q(x) is < 3 then the conjunction is false so the first part of implication is false that means even if second part is true, implication is still false. And if its false than this statement isnt the law of unary calculus of quantifiers (Im not sure if thats right translation, english isn't my first language). But I guess any way of proving true or false is fine. In term of my efforts I dont know how to approach this task at all. I get what I have to do but I got no clue how to find the x that proofs that statement is true or false. Again im psychology student at the university and last time I saw math was 7 years ago and it was a lot simplier than this. Thank you all for replies.

 

[brainoverload]

Here is the example. Forgot to attach it in previous post.

 

[stapel]

I dont think there is domain of discourse.... But I guess any way of proving true or false is fine. In term of my efforts I dont know how to approach this task at all.

Has this topic not yet been covered in your textbook nor in the classroom lecture...? Because we don't do homework here, and we're really not set up to provide days or weeks of course instruction.