584 Appendix A • Logic and Proofssentences may express the same proposition. For example, "John is five feet
eleven inches tall" is the same proposition as "If John were one inch taller
he would be exactly six feet tall." Moreover, translating the sentence into a
different language would not make it a different proposition.
Thus, the notion of a proposition is really an abstraction. The words we
use are merely a tangible representation of the proposition. T hey bear the
same relationship to the (abstract) proposition as a tangible triangle drawn on
paper or chalkboard bears to the (abstract) triangle it represents. In fact, all
mathematical concepts are abstractions!
C OMPOUND PROPOSITIONSSimple propositions are often combined together to make compound propo-
sitions. Logicians have identified five log ica l connectives commonly used for
this purpose:
THE F IVE LOGICAL CONNECTIVES
Connective N a m e Example Symbolic form
and conjunction Pand Q PA Q
or disjunction Por Q PV Q
not negation not P rv p
if ... then implication if P, then Q P-=;. Q
if and only if bi-implication P if and only if Q p {::} QIn the study of logic we analyze the relationship between the truth-values
of simple propositions and the truth-values of related compound propositions.
D e finition A.1.2 The conjunction "P and Q" is symbolized PA Q and is
defined by the truth-table:
Table A .1p Q P A Q
T T T
T F F
F T F
F F FTable A. l shows that P A Q is true only when P and Q are both true; it
is false in all other cases.