Respuesta :
The mistake lies in steps 3 and 5.
In fact, you know that there exists some element that satisfies p(x), and some element that satisfies q(x), but you can't assume that the same element satisfies p(x) and q(x) at the same time.
So, you are only allowed to say
p(c) existential instantiation from (2)
q(d) existential instantiation from (4)
and so you can't do the conjunction in step 6 anymore.
Here's an example: if you follow this logic, you would say something like:
"There exists a person who is born in Italy and there exists a person who is born in the USA. Therefore, there exists a person who is born in both Italy and the USA"
which is clearly false: altought there exists a person for each proposition, they are not the same person.
This is even easier if q is the negation of p:
"There exists a number greater than 10, and there exists a number lesser than 10. Therefore, there exists a number that is greater than and lesser than 10 at the same time"
which is clearly absurd
Here we want to look at the steps of a demonstration, and see where we have errors. We will see that the errors are in steps 3 and 5.
"∃x p(x) ∧ ∃x q(x) is true" means that:
"there exists, at least, a value of x such that p(x) is true and there exists, at least, a value of x such that q(x) is true"
Notice that these two values of x can be equal, but not necessarily.
While the second statement:
"∃x(p (x) ∧ q(x)) is true"
means that:
"There exists, at least, a value of x such that p(x) and q(x) is true"
In this case, we have a single value of x for both propositions.
So already we can see that the statements are not equivalent.
Now let's see why.
If we look at the demonstration, we can see that in steeps 3 and 5 we have:
3) p(c) existential instantiation from (2)
5) q(c) existential instantiation from (4)
Where (2) and (4) are:
2) ∃x p(x) simplification from (1)
(There exists a value of x such that p(x) is true)
4) ∃x q(x) simplification from (1)
(There exists a value of x such that q(x) is true)
Now, again, if you look and (2) and (4), you know that these values of x do exists, but there is nothing that tells you that these values of x are the same values.
Then you can't say that:
3) p(c) existential instantiation from (2)
5) q(c) existential instantiation from (4)
This implies that both propositions are true for the same value x = c.
Thus, the errors are in these two steps.
If you want to learn more, you can read:
https://brainly.com/question/13011669
