Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I want to interpret the below sentences...

  1. ∀x∃y(Cube(x) → (Tet(y) ∧ LeftOf(x, y)))
  2. ∃y∀x (Cube(x) → (Tet(y) ∧ LeftOf(x, y)))

Actually I could interpret the first sentence.

Meaning: Every Cube is left of some Tet

But.. I can't interpret the second sentence.

Let (Cube(x) → (Tet(y) ∧ LeftOf(x, y))) be P(x, y).

There exits some y such that for all x P(x, y) is ture?

.. I don't know.......

please help me to know about it!!

share|improve this question

2 Answers 2

Take it one step at a time. You have $$\exists y\forall x\left(\operatorname{Cube}(x)\to\Big(\operatorname{Tet}(y)\land\operatorname{LeftOf}(x,y)\Big)\right)\;.\tag{1}$$ The first step in translating it is straightforward: just expand the quantifiers and predicates into words.

(a) There is a $y$ such that for every $x$, if $x$ is a cube, then $y$ is a tetrahedron and $x$ is to the left of $y$.

Now notice that if there is at least one cube, the statement y is a tetrahedron does not depend in any way on $x$. We could just as well say:

(b) There is a $y$ such that $y$ is a tetrahedron, and for every $x$, if $x$ is a cube, then $x$ is to the left of $y$.

It would mean the same thing.

The qualifier if there is at least one cube is necessary, because if there are no cubes, $(1)$ and (a) are vacuously true. To keep matters simple, I’ll assume for the moment that there is at least one cube and derive a simple English equivalent; once we have that, we can modify it to get rid of the extra assumption.

Now we can start turning (b) into more ordinary English:

(c) There is a tetrahedron such that every cube is to the left of that tetrahedron.

Or in even more straightforward English:

(d) There is a tetrahedron that is to the right of every cube.

Remember, though, this was equivalent to $(1)$ and (a) only on the extra assumption that there is at least one cube, and that $(1)$ and (a) are vacuously true if there are no cubes. To make (d) equivalent to $(a)$, we must add an alternative:

(e) Either there are no cubes, or there is a tetrahedron that is to the right of every cube.

share|improve this answer
    
Wow.. it's really detail :D. Thank you! –  Jonghwan Hyeon May 15 '12 at 18:21
    
This is not quite right. Your (a) and (b) are only equivalent under the additional assumption that there is a cube, that is, $\exists x\, \text{Cube}(x)$. If there are no cubes, then there is no requirement that $y$ be a tetrahedron, since the implication is vacuously satisfied. –  Alex Kruckman May 15 '12 at 19:39
    
Hmm.. sorry.. but I don't know the difference between first and second sentence.. I think the both have same meaning, although their english sentences are different.. is that right? –  Jonghwan Hyeon May 15 '12 at 23:12
    
@JonghwanHyeon: Not quite. Suppose that there are no cubes and no tetrahedrons; then (a) is vacuously true, and (b) is false. –  Brian M. Scott May 16 '12 at 3:20
    
@Brian M. Scott: Oh I mean, between ∀x∃y(Cube(x) → (Tet(y) ∧ LeftOf(x, y))), and ∃y∀x (Cube(x) → (Tet(y) ∧LeftOf(x, y))) –  Jonghwan Hyeon May 16 '12 at 3:23

The exact paraphrase of the second sentence would be: there exists y such that for all x if x is a cube than y has property tet and it holds that leftof(x,y). Also your abbreviation seems correct.

share|improve this answer
    
thank you for your answer :D –  Jonghwan Hyeon May 15 '12 at 18:20

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.