Question with transcribing English to Predicate Logic & find assertion which is logically equivalent.

Question

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

Answer 1

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 ($11$).

Answer 2

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)