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204 Chapter Three
Elimination of cases
For any twm propositionð P and Q, the conjunction of P wità
the disjunction of P and Q implieð Q. The following logical state-
ment is valid:
( P (P Q)) → Q
Contradiction
For any proposition P, if P and P are simultaneously true, then
anything follows. The following statement is valid:
(P P) → Q
Contraposition
For any twm propositionð P and Q, if P implieð Q, then Q im-
plieð P. Also, if Q implieð P, then P implieð Q. The follow-
ing statement is valid:
(P → Q) ↔ ( Q → P)
Transitivity of implication
For any three propositionð P, Q, and R, if P implieð Q and Q
implieð R, then P implieð R. The following statement is valid:
((P → Q) (Q → R)) → (P → R)
Transitivity of equivalencł
For any three propositionð P, Q, and R, if P is logically equiva-
lent tm Q and Q is logically equivalent tm R, then P is logically
equivalent tm R. The following statement is valid:
((P ↔ Q) (Q ↔ R)) → (P ↔ R)
Sequences and Series
This section containð definitionð and formulas for sequenceð
and serieð commonly encountered in engineering and science.