PostItNotes
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- Mar 23, 2007
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I'm taking a course in mathematical logic and I am currently working on a few proofs.
1. Prove or disprove: {A V B, not B} "models" A
(I don't know how to make the "models" symbol. It looks like a vertical line directly in front of an equal sign.) Do I use the satisfaction relation or the consequence relation on this? Do I need some type of interpretation? I don't understand what to do with the two parts in the braces. So far, we've only dealt with either (A V B) or (not B) alone. We haven't dealt with sets of formulas in braces and the book doesn't give very many examples. We're using the book Mathematical Logic by Ebbinghaus, Flum, and Thomas if that helps.
2. Prove or disprove: "there exists"xP(x) "models" "for all"xP(x) where P is a unary relation symbol.
I'm thinking that this is false, but I don't know how to prove it mathematically. I'm thinking it's false because I also have to prove or disprove the reverse and I'm thinking one has to be true and one false. I just don't know, though.
3. Once again, P is a unary relation symbol. I have to determine if these structures are isomorphic and I also have to prove or disprove my findings.
a. Is (Z, evens) isomorphic to (Z, odds)?
b. Is (Z, odds) isomorphic to (N, evens)?
c. Is ({5,6,7,8},{5,6}) isomorphic to ({5,6,7,8},{5,6,7})?
I'm thinking that a. is an isomorphism since the integers are isomorphic to themselves and the evens have the same number of elements as the odds.
I'm thinking that b. is an isomorphism since the integers and the natural numbers are both infinite sets and for the same even/odd reason.
I'm thinking that c. is not an isomorphism because the sets are the same, however, the relations {5,6} and {5,6,7} are not the same size.
The main problem is that I don't know how to write this stuff out mathematically.
Thanks in advance, guys!
1. Prove or disprove: {A V B, not B} "models" A
(I don't know how to make the "models" symbol. It looks like a vertical line directly in front of an equal sign.) Do I use the satisfaction relation or the consequence relation on this? Do I need some type of interpretation? I don't understand what to do with the two parts in the braces. So far, we've only dealt with either (A V B) or (not B) alone. We haven't dealt with sets of formulas in braces and the book doesn't give very many examples. We're using the book Mathematical Logic by Ebbinghaus, Flum, and Thomas if that helps.
2. Prove or disprove: "there exists"xP(x) "models" "for all"xP(x) where P is a unary relation symbol.
I'm thinking that this is false, but I don't know how to prove it mathematically. I'm thinking it's false because I also have to prove or disprove the reverse and I'm thinking one has to be true and one false. I just don't know, though.
3. Once again, P is a unary relation symbol. I have to determine if these structures are isomorphic and I also have to prove or disprove my findings.
a. Is (Z, evens) isomorphic to (Z, odds)?
b. Is (Z, odds) isomorphic to (N, evens)?
c. Is ({5,6,7,8},{5,6}) isomorphic to ({5,6,7,8},{5,6,7})?
I'm thinking that a. is an isomorphism since the integers are isomorphic to themselves and the evens have the same number of elements as the odds.
I'm thinking that b. is an isomorphism since the integers and the natural numbers are both infinite sets and for the same even/odd reason.
I'm thinking that c. is not an isomorphism because the sets are the same, however, the relations {5,6} and {5,6,7} are not the same size.
The main problem is that I don't know how to write this stuff out mathematically.
Thanks in advance, guys!