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26 The Nuts and Bolts of Proof, Third Edition
Quantifiers are not the only possible source of problems when
constructing the negation of a statement. The logical connectors "or" and
"and" have to be handled carefully as well.
The composite statement "C or D" is true when either one of the
statements C or D is true. While it is possible for both statements to be true,
it is not required. Unless otherwise indicated, the "or" used in mathematics is
inclusive; that is, it includes the possibihty that both parts of the statement
are true. This use of "or" is different from its everyday use, when "or"
suggests a choice between two possibiHties (as in, "Would you like to have
coffee or tea?"); therefore, for the statement "C or D" to become false, both C
and D must be false. Thus, the negation of "C or D" (i.e., the statement "not
'C or D'") is the statement "'not C and 'not D.'"
The composite statement "C and D" is true when both statements C and
D are true. Therefore, for it to become false, it is sufficient that either C or D
is false; thus, the negation of "C and D" (i.e., the statement "not 'C and D' ") is
the statement " 'not C or 'not D.' "
The truth tables can reinforce and clarify what has just been stated in the
previous paragraphs:
Following is the truth table for the negative of the statement "C or D" (i.e.,
"not 'C or D'").
c
D
C or D Not' "C or D"
T
T
T
F
T
F
T
F
F
T
T
F
F
F
F
T
Following is the truth table for the statement " 'not C and 'not D.''
C D Note NotD "Not C and "Not D"
T T F F F
T F F T F
F T T F F
F F T T T
By comparing these two tables, we can conclude that the statements "not
'C or D'" and " 'not C and 'not D'" are indeed logically equivalent. The
constructions of the truth tables for the statements "not 'C and D'" and
" 'not C or 'not D'" is left as an exercise.