to the set
2. Statement: (∀x ∈
For each x ∃
Negation: (∃x ∈
There exists x ∈
3. Statement: (∀x ∈
For each x ∈
Negation: (∃x ∈
Remark. The third example is more complicated than the other two. Hence we offer a step-by-step analysis of the process of negation.
In Example 3, we negate the following statement.
(∀x ∈
That is, we produce a statement that is logically equivalent to the following.
¬((∀x ∈
For our first step, we change the quantifiers at the beginning of the sentence and move the symbol ¬ to their right.
(∃x ∈
Notice that the statement governed by the two initial quantifiers has the form ¬(p ⟶ q). Since ¬(p ⟶ q) is logically equivalent to p ∧ ¬q, we obtain the following sentence.
(∃x ∈
Now we transform the sentence ¬(∃z ∈
(∃x ∈
Since x > z > y is shorthand for (x > z) ∧ (z > y), we have the following sentence.
(∃x ∈
Since ¬(p ∧ q) is logically equivalent to ¬p ∧ ¬q, we transform the sentence as follows.
(∃x ∈
Finally, since ¬(a > b) can be written more simply as a ≤ b, we get the sentence below.
(∃x ∈
Remark. Every step in simplifying a negation moves the symbol ¬ further to the right. When all the ¬ symbols are as far to the right as possible, we have simplified the negation as much as possible.
Exercises (1.7) Classify each