Why Read This Book? - Index of

Solution
1.4 Negations of Statements 37
1. There exists x ∈ Z such that x2 < 0.
2. There exists x ∈ Z such that x is divisible by 6 but not divisible by 3.
3. There exists a real number that is neither rational nor irrational. �
Warning: Universal statements are often written in if-then form with the universal
quantifier omitted. This is perfectly acceptable in mathematical discourse,
but the presence of the universal quantifier must be understood if you are going
to negate the statement correctly.
Example 1.4.9 A mathematical statement such as
If x>1, then x 3 − x>0.
is an acceptable simplification of the more formal
For all x,ifx ∈ R and x>1, then x 3 − x>0.
even though it contains the indeterminate x and does not specify that the context
is the real numbers. However, the formalized version must be understood in order
to construct the correct negation:
There exists a real number x such that x>1 and x 3 − x ≤ 0. �
EXERCISE 1.4.10 State in words a negation for each of the following statements.
(a) If n is a natural number and n ≥ 5, then n2 < 2n .
(b) The only solutions to the equation x2 − x = 0 are nonnegative.
(c) On a clear day, you can see forever.
(d) If −1 ≤ x ≤ 10, then x2 ≤ 100.
(e) Every time a bell rings, an angel gets his wings.
(f) No one in the class made an A on the final.
1.4.4 Negations of Statements with the Existential Quantifier
Now we negate statements involving the existential quantifier. Consider the
following.
Someone traveling in my car didn’t chip in enough money for gas.