Mathematical Writing - Vivaldi Franco 2014
Negating Logical Expressions
Mathematical Sentences
If is a logical expression, then its negation is , which is false if is true and vice-versa. Relational expressions are negated by crossing out the operator with a forward slash—see (4.3) and (4.5), Sect. 4.1. The only exceptions are the inequalities, whose negations have dedicated symbols.
Theorem 4.1 provides the negation formulae for the main compound expressions
(4.26)
The first two formulae are items (i) and (ii) of the theorem (de Morgan’s laws); the third follows immediately from (iii) and (i). Note that the negation of an implication does not have the form of an implication.
If is an expression with quantifiers and boolean operators, then the expression unfolds into an equivalent expression which we seek to determine. The following sentences contain a negation and a universal quantifier:
Not all polynomials have real roots. A square matrix is not necessarily invertible.
We recognise that these are disguised existence statements:
There is a polynomial with no real roots. There is a square matrix which is not invertible.
The following theorem confirms this observation.
Theorem 4.3
Let be a predicate on a set , and let
Then
(4.27)
PROOF. From (4.21), we have the boolean equivalence
where is the image of under (see Sect. 2.2). But then
Using again (4.21), we obtain
as claimed.
The second formula is proved similarly—see exercises.
It’s now easy to deduce the rule for negating expressions with two or more quantifiers. Let be a predicate on , and let us consider the expression
Repeated applications of Theorem 4.3 give
It should be clear how similar formulae may be derived, having any combination of two quantifiers.
Finally, we consider the case of an arbitrary sequence of quantifiers, e.g.,
where the are sets, and is a predicate on the cartesian product . Parentheses make it clear that this is a nested array of predicates:
Repeatedly applying Theorem 4.3, from the outside to the inside, we obtain
Namely, the negation of is obtained by replacing each with , and vice-versa, without changing their order, and then replacing with . A formal proof requires the principle of induction (see Chap. 8).
EXAMPLE. The statement
(4.28)
is false (why?). We negate it using Theorem 4.3, obtaining a true statement:
EXAMPLE. Lagrange’s theorem, expressed symbolically in Eq. (4.25), would be false if four squares were replaced by three squares. This fact is written symbolically as
In words:
There is a natural number which cannot be written as the sum of three squares.