I am doing a HW assignment for First Order Logic with english sentences and this is one of the questions. Not exactly sure of how to approach it and to answer it.
Q1. [10] Decide each sentence is valid (necessarily true). Justify your answer.
1)$(\exists x (x = x)) \rightarrow (\forall y \exists z (y = z))$
2) $\forall x (\mathrm{Smart}(x) \vee (x = x))$
I don't really know how to think about these questions, I feel like question one is saying that there exists for example a person x Therefore all of people y are the same as this one person z. Which I feel like it could be wrong.
I also feel like questions two is true but only because Person X is the same as Person X
Hint
For 1), the antecedent says "There is an $x$ such that $(x=x)$", that amounts to : "there is something that is equal to itself".
The standard interpretation of FOL uses non-empty domain, that means that it is always true that there is something, and thus that there is something that is equal to itself.
Regarding the consequent, it says "For every $y$ There is a $z$ such that $(y=z)$".
And also this is true, because for every object there is soemthing equal to it: at least the object itself.
Conclusion: both sub-formulas are always True, and "if True, than True", is True.
The same approach for 2) : "For every $x$, either $x$ is Smart [maybe it is smart, maybe not] or $x$ is equal to itself [for sure it is true, whatever $x$ is]".