# Negation of Quantified Statements

I was wondering if someone could walk me through how to do this type of question, my teacher didn't really explain it well enough for me to follow along.

QUESTION: Let D = E = {−3, 0, 3, 7}. Write negations for each of the following statements and determine which is true, the given statement or its negation. Explain your answer.

(i) ∀x ∈ D, ∃y ∈ E such that x + y = 0.

(ii) ∃x ∈ D such that ∀y ∈ E, x + y = y.

(iii) ∀x ∈ D, ∃y ∈ E such that xy ≥ y.

(iv) ∃x ∈ D such that ∀y ∈ E, x ≤ y.

• The trick is to rewrite $\neg\forall x\in A\phi (x)$ as $\exists x\in A\neg\phi (x)$ and $\neg\exists x\in A\phi (x)$ as $\forall x\in A\neg\phi (x)$.
– J.G.
Commented Feb 5, 2019 at 7:55

Hint

(i) $$∀x ∈ D \ ∃y ∈ E \ (x + y = 0)$$.

Consider the expression $$(x + y = 0)$$ : it expresses a "condition" on $$x$$ and $$y$$.

We have to "test" it for values in $$D = E = \{ −3, 0, 3, 7 \}$$, and specifically we have to check if :

for each number $$x$$ in $$D$$ there is a number $$y$$ in $$E$$ (which il the same as $$D$$) such that the condition holds (it is satisfied).

The values in $$D$$ are only four : thus it is easy to check them all.

For $$x=-3$$ we can choose $$y=3$$ and $$x+y=0$$ will hold.

The same for $$x=0$$ and $$x=3$$.

For $$x=7$$, instead, there is no way to choose a value for $$y$$ in $$E$$ such that $$7+y=0$$.

In conclusion, it is not true that : for each number $$x$$ in $$D$$ ...

Having proved that the above sentence is FALSE, we can conclude that its negation is TRUE.

To express the negation of a quantified formula, we have to consider that $$\lnot \forall$$ is the same as $$\exists \lnot$$ and, in turn, that $$\lnot \exists$$ is the same as $$\forall \lnot$$.

Thus, the negation of (i) will be : $$\lnot [∀x ∈ D \ ∃y ∈ E \ (x + y = 0)]$$, i.e.

$$∃x ∈ D \ ∀y ∈ E \ (x + y \ne 0)$$.

Final check; the new formula expresses the fact that :

there is an $$x$$ in $$D$$ such that, for every $$y$$ in $$E$$ it is not true that $$(x+y=0)$$,

and this is exactly what we have found above with $$7$$.

Negation of (i):
exists x in D such that for all y in E, x + y /= 0.