Why Read This Book? - Index of

36 Chapter 1 Language and Mathematics
1.4.3 Negations of Statements with the Universal Quantifier
Suppose someone makes the following claim.
Every person in the class passed the first exam.
For you to deny this claim, what sort of statement would you make? 6 You would
probably say something like
At least one person in the class failed the first exam.
This is where the most formal version of the universal statement comes in handy.
Let C be the set of students in the class and let P be the set of students in the class
who passed the first exam. We can write the original statement as
(∀x)(x ∈ C → x ∈ P) (1.19)
and its negation would be
(∃x)(x ∈ C ∧ x/∈ P)
Notice in the negation that (x ∈ C ∧ x/∈ P) is a negation of the statement (x ∈
C → x ∈ P). Using the property notation P(x), we can therefore say the following.
¬[(∀x)(P(x))] ⇔(∃x)(¬P(x)) (1.20)
So the trick to negating a universal statement is that the ¬ symbol crawls over the
(∀x) and converts it to (∃x) as it goes.
Another symbolic rendering of statement (1.19) would be
(∀x ∈ C)(x ∈ P)
whose negation would be
(∃x ∈ C)(x /∈ P)
Notice that the (∀x ∈ C) becomes (∃x ∈ C) as the ¬ symbol crawls over it, then
the statement (x ∈ P) is negated.
Example 1.4.8 State in words a negation of the following statements.
1. For all x ∈ Z, x2 ≥ 0.
2. For all x ∈ Z,ifx is divisible by 6, then x is divisible by 3.
3. Every real number is either rational or irrational.
6 One slacker is all it takes.