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October 3

Vertical bars in set theory

In this problem:

(2) Find a model 𝒥 for the following set of sentences, with |𝒥| = {1, 2, 3, 4, 5}:

1. ∀x∃yRxy

2. 𝑓(x, 𝑠(x)) = sum(1, prod(x, 2))

What do the vertical bars around the second instance of 𝒥 mean? Is it saying that {1, 2, 3, 4, 5} is the set of sentences that 𝒥 is supposed to model?

MonroeL (talk) 04:27, 3 October 2017 (UTC)[reply]

I think it's probably using |𝒥| to mean the universe of the model (sometimes called the domain, the underlying set, or the carrier). That is, your "objects" are the numbers 1 through 5.

~~(Note that I don't think this is very standard notation. More usually, |𝒥| would be the cardinality of the universe.)~~ I take it back — actually I think this is reasonably standard.

I don't think there's enough information here to give a sensible answer to the problem. Are we supposed to take "sum" and "product" to have their usual meanings on 1 through 5? That doesn't make a lot of sense, given that some of the answers would not be objects of the model. So I'm not quite sure what's going on. --Trovatore (talk) 04:40, 3 October 2017 (UTC)[reply]

Thanks for the speedy response!

In this problem, the function s is the successor function, and sum and prod are the usual meanings for the terms "sum" and "product."

I just saw this information from the instructor: "you need to find a relation over |𝒥| that makes 1. true, and a function over |𝒥| that makes 2. true." I don't think that's communicated very clearly in the original problem, but it's alright!

I appreciate the help a lot. Have a great day! MonroeL (talk) 05:22, 3 October 2017 (UTC)[reply]

It's still very peculiar. For example, in the second clause, take x to be 3. Then you have 𝑓(x, 𝑠(x)) = 𝑓(3, 4) = sum(1, prod(3, 2)) = 7, if I'm following so far. But 7 is not an element of the universe of the model, so as far as I can tell this doesn't make sense. --Trovatore (talk) 05:41, 3 October 2017 (UTC)[reply]

Side comment: possibly, if the question gets answered, the answer could be added to the List of mathematical symbols...? --CiaPan (talk) 06:09, 3 October 2017 (UTC)[reply]