1549901369-Elements_of_Real_Analysis__Denlinger_

A.2 The Logic of Predicates and Quantifiers 603

Theorem A.2.15 Negations of Categorical Propositions:

(a) '""UP =EN

(b) '""EP =UN

(c) '""UN=: EP

(d) '""EN =UP

Proof: (a) '""UP = '""Vx, [A(x) => B(x)]

= 3x 3 '""[A(x) => B(x)]

= 3x 3 [A(x)/\'"" B(x)] EN.

(b) '""EP = '""(3 x 3 [A(x) /\ B(x)])

= Vx, '""[A(x) /\ B(x)]

=: Vx, [rv A(x)V '""B(x)]

Vx, [A(x) => '""B(x)] = UN.

(c) Using (b), we have'"" UN = '""(rv EP) EP.

(c) Using (a), we have'"" EN = '""(rv UP) UP. •

Negation of the categorical propositions is easy to remember using the
following rectangular diagram, where the diagonals represent negations:

UXP EP

U EN

Examples A.2.16 Negations of categorical propositions:

0

(a) rv(All eagles are graceful) = Some eagles are not graceful.

(b) rv(Someone stole my wallet) =No one stole my wallet.

(c) rv(No one can jump that high) =Someone can jump that high.

(d) rv(Someone will not pass the exam.) = Everyone will pass the exam.

MULTIPLY QUANTIFIED STATEMENTS

It is quite common in mathematics for statements to contain more than
one quantified variable. When translating such statements into logical sym-
bolism, great care must be taken to capture the precise meaning intended by