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Given these conditions... $P(x) = x$ is a cow, $Q(x) = x$ makes milk, $R(x,y) =$ both $x$ and $y$ are the same object.

This expression says the following.. $$(\exists x)[P(x) \wedge Q(x)]$$ and another $$(\exists x)[P(x)] \wedge (\exists x)[Q(x)]$$

I translated those to the following... 1. Some cows produce milk. 2. There are some cows that produce milk.

Are these correct?

Bram28
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MD_90
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  • Your $R(x,y)$ is never used in your question. Is this correct? – Rory Daulton Feb 01 '15 at 20:58
  • Second one is wrong. It should be - There are some cows, and there are some creatures that produce milk. You cannot infer from this logical statement that those creatures are indeed cows. – barak manos Feb 01 '15 at 21:04
  • Thank you Dillion I'm new to this – MD_90 Feb 01 '15 at 21:16
  • @MD_90 What do you mean the course doesn't allow for that symbol? The symbol $\exists !$? Do you know what it means? – Daniel W. Farlow Feb 01 '15 at 21:16
  • Sadly I don't @induktio :( the symbol was not covered in class therefore they don't allow us to use it in the expression notation – MD_90 Feb 01 '15 at 21:18
  • @MD_90 No worries. You're not really supposed to use it (I was just trying to interact with Rory who definitely knows what $\exists !$ means). It is the unique existential quantifier. Check out this post to learn more about it: http://math.stackexchange.com/questions/1119836/logic-how-to-write-exists-x-without-the-exists-symbol/1119847#1119847 – Daniel W. Farlow Feb 01 '15 at 21:20
  • @MD_90 I am actually not a fan of Rory's answer or what barak manos said: I feel there's an important difference between plurality and singularity with the existential quantifier. The existential quantifier guarantees what? Existence. Not of some this or that but one. At least that's how I view it anyway. – Daniel W. Farlow Feb 01 '15 at 21:23
  • Is there any good sites or ways to make these easier to figure out so translating between the two is simpler? I know the course covers propositional logic, graph theory(large emphasis on this topic), algorithm analysis, and induction – MD_90 Feb 01 '15 at 21:24

2 Answers2

2

I would answer it as follows:

$$(\exists x)[P(x)\land Q(x)]$$

means, in the context of your statements, that (note this is a very strict interpretation) there exists a cow that makes milk (maybe not more than one but perhaps).

Now,

$$(\exists)[P(x)]\land (\exists)[Q(x)]$$

means that there exists a cow and there exists a milk maker.

Daniel W. Farlow
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1

The first is basically correct, though I would prefer "A cow makes milk." Your formulation implies that there are multiple cows, i.e. at least two cows, that make milk, while the logic statement says at least one cow. So I prefer my formulation. Perhaps "There is at least one cow that makes milk" is the most clear, though somewhat clumsy.

For (2): There is a cow and something makes milk.

Rory Daulton
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  • Is the first one really correct though? Some cows is a lot different than a cow, of course. Nitpicky, but I think it's more than a matter of preference in this context. – Daniel W. Farlow Feb 01 '15 at 20:59
  • @induktio: It is largely a matter of preference. But the OP's formulation implies that there are multiple cows, i.e. at least two cows, that make milk, while the logic statement says at least one cow. That's why I prefer my formulation. I have edited my answer to make that more clear. – Rory Daulton Feb 01 '15 at 21:00
  • Interesting. I guess I take the $\exists$ to be very literal in the sense that it does not guarantee more than one cow (even though it obviously shouldn't be interpreted as $\exists !$). – Daniel W. Farlow Feb 01 '15 at 21:03
  • I'm currently taking a Discrete Mathematics course for my computer science major. It's hard for me to see how to translate between the expression form, and English form. I am trying though but any insight that will help makes these easier would be very helpful :) – MD_90 Feb 01 '15 at 21:13
  • course doesn't allow that symbol @induktio – MD_90 Feb 01 '15 at 21:14
  • @RoryDaulton can you suggest any good ways of approaching these problems in both forms and any good reads that help practice and show good examples of both? I could sure benefit from that. I'm a cs major with a mathematics minor and in order to be the best I can be at both I got to work hard to master the material of both worlds. Same goes for induktio. :) – MD_90 Feb 01 '15 at 22:08