Wikipedia:Reference desk/Archives/Mathematics/2018 September 26
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September 26[edit]
Mathematics for Computer Science - Propositional Functions[edit]
Suppose the domain of the propositional function P(x,y) consists of pairs x and y, where x = 1, 2, or 3, and y = 1, 2, or 3. Write out the following proposition using disjunctions and conjunctions only:
∃xP(x,y)
I don't even know where to start... (x ∧ y) ∨ (y ∧ x) type thing is what the question is asking for? Because I don't see what that would mean anyway, so any help would be greatly appreciated.
Kidoooo$ (talk) 00:21, 26 September 2018 (UTC)
- I think the kind of expression they have in mind is something like
- P(1,1) ∨ (P(1,3) ∧ P(2,3))
- That's not the answer, but it's in the general form the answer would take so hopefully that will get you started.
- I'm a bit confused though since ∃xP(x,y) has a free variable y, so it defines a propositional function rather than a proposition. Either there is a typo somewhere, or your text/prof is using nonstandard terminology, or the terminology I'm used to is nonstandard (or a combination of these). So maybe the answer is supposed to involve expressions like P(2,y). --RDBury (talk) 05:42, 26 September 2018 (UTC)
I appreciate all of your answers and comments that are helping me understand the material. I also wanted to know if the way I did this is valid or correct:
Suppose the domain of the propositional function P(x, y) consists of pairs x and y, where x = 1, 2, or 3, and y = 1, 2, or 3. Write out the propositions below using disjunctions and conjunctions only.
∃x∀y¬P(x, y)
The above is equivalent to ¬(∀x∃yP(x,y)) So can I write (∀x∃yP(x,y)) using disjunctions and conjunctions and then further adjust it with the negation?
¬(∀x∃yP(x,y)) ≡ ¬((P(1,1) ∨ P(1,2) ∨ P(1,3)) ∧ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∧ (P(3,1) ∨ P(3,2) ∨ P(3,3))) ≡ ¬(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∨ ¬(P(2,1) ∨ P(2,2) ∨ P(2,3)) ∨ ¬(P(3,1) ∨ P(3,2) ∨ P(3,3)) ≡ (¬P(1,1) ∧ ¬P(1,2) ∧ ¬P(1,3)) ∨ (¬P(2,1) ∧ ¬P(2,2) ∧ ¬P(2,3)) ∨ (¬P(3,1) ∧ ¬P(3,2) ∧ ¬P(3,3))
Is that a valid method and correct answer? Thanks in advance. Kidoooo$ (talk) 02:41, 27 September 2018 (UTC)