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Anyone knows how to solve the question 10(especially 10(b)) & 11?

I actually have the standard/model answers, but I don't really know how they understand the question and solve it step by step.

Anyone can explain it and help me out....I am in trouble...

I have answers already...I just want the detailed explanation.

Question 10 & 11

Standard/Model Answer

Joseph Tam
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2 Answers2

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For ($10$b), I would cast it as: $$\forall x\forall y\,{\big[}P(x,x,y) \rightarrow \bigl(G(y,0)\lor E(y,0)\bigr){\big ]}$$

The thought process:

$y = x^2\;$can't happen unless $y \ge 0$.

Hence, if $y=x^2$, we must have $y \ge 0$.

Using if-then:$\;$If $y=x^2$, then $y \ge 0$.

Converting to symbols:$\;P(x,x,y) \rightarrow \bigl(G(y,0) \lor E(y,0)\bigr)$.

Finally, quantify the variables $x,y:\;\forall x \forall y\,{\big[}P(x,x,y) \rightarrow \bigl(G(y,0)\lor E(y,0)\bigr){\big ]}$.

@Fabio Somenzi answered problem ($11$).

quasi
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  • I edited it with posting the model answer......regarding to your answer, it's correct..logically...so thanks, but why would the model answer of 10(b) comes with 3 steps, do you know how they figure out?? Any why they put the quantifier in front and why you don't?...........so confusing.. – Joseph Tam Oct 08 '17 at 04:30
  • I think they'e trying to transform the word version, one small piece at a time. I see that as more confusing than helpful. – quasi Oct 08 '17 at 04:35
  • Can I also solve the problem like this?? ∀y[∀xP(x,x,y)→(G(y,0)∨E(y,0)] – Joseph Tam Oct 08 '17 at 04:40
  • As far as missing quantifiers, I've edited them in. They should be there. In an informal context, for symbols that are viewed as variables (non-constants), the universal quantifiers can be regarded as implied. – quasi Oct 08 '17 at 04:41
  • Yes, the version in your previous comment is essentially the same. – quasi Oct 08 '17 at 04:43
  • Just one more question, for the standard/model answer of 10(b), there are 3 step as indicated in there. How would you explain the step 1?? You answer is much more clear, but I still want to know how the step 1 comes out... – Joseph Tam Oct 08 '17 at 06:08
  • In words, step $1$ says: For all integers $y$, there is no integer $x$ such that $y=x^2$ and $y$ is not greater or equal to zero. But such steps are not cast in stone, so I wouldn't take those exact steps too seriously (unless the teacher requires it). – quasi Oct 08 '17 at 06:18
  • Thanks bro...it helps a lot on my university study!! Math professor doesn't teach well...every student agrees on a thing..."That professor sucks"

    lol.....Anyway, you absolutely deserve that "Best Answer" :D (I wanna share the joy with the another answerer "Fabio Somenzi" but the system doesn't allow....what a shame :p)

     – Joseph Tam Oct 08 '17 at 07:19
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For Problem 11, you want to express that there is a unique $x$ that satisfies $P$. It can be done as follows:

$$ \exists x(P(x) \wedge \forall y (P(y) \rightarrow Q(x,y))) \enspace.$$

It says that there is an $x$ that satisfies $P$, and for all $y$, if they satisfy $P$, they must be equal to $x$.

@quasi answered 10(b)

Fabio Somenzi
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