I want to convert these statements into symbolic form and then negate them:
(a) For all integers x there is an integer y that is smaller than x.
(b) Every integer greater than 4 is odd.
(c) There is an integer less than 10 that belongs to all elements of P(Z).
This I what I think the symbolic forms are:
(a) $∀x ∈ Z, ∃y ∈ Z, y < x$
(b) $∀x ∈ Z, x > 4, odd(x)$
(c) $∃x ∈ Z, x < 10, x ∈ P(Z)$
With the negations:
(a) $∃x ∈ Z, ∀y ∈ Z, y≥x$
(b) $∃x ∈ Z, x ≤ 4, even(x)$
(c) $∀x ∈ Z, x ≥ 10, x ∉ P(Z)$
(b) and (c) are particularly confusing me and I think I have made a mistake in either/both the symbolic form or negation. Did I do these correctly and if not can someone explain where I went wrong?
b) should be $\forall x \in Z , x > 4 \to Odd(x)$
Sentence c) is actually ambiguous, and can be interpreted in two different ways:
One interpretation is to say that there is a specific number smaller that 10 that is an element of all the elements of $P(Z)$. That is: it is the same number every time for all those elements. This would be symbolized as:
$\exists x \in Z , \forall y \in P(Z), x < 10 \land x \in y$
So, this claim would be true if, for example, we find the number $6$ to be an element of every element of $P(Z)$.... which is the same as saying that $6$ would be an element of every subset of $Z$ ... which is clearly not true.
A different interpretation is to say that all elements of $P(Z)$ have their own number smaller than $10$ that's in the set, but it need not be the same number for all elements.
This would be symbolized as:
$\forall y \in P(Z), \exists x \in Z , x < 10 \land x \in y$
Note that also this claim is a false claim: the set containing of all numbers greater than $10$, for example, is a subset of $Z$, and hence an element of $P(Z)$, but it has no number than smaller than $10$ in it. Nor does the empty set.
So .. which on is meant? Well, we just don't know. That's the difference between English and logic: English is ambiguous, but in logic we don't have such ambiguities.