1549901369-Elements_of_Real_Analysis__Denlinger_

594 Appendix A • Logic and Proofs

Table A.10

p Q R QVR p /\ p /\ Q p /\ R (P /\ Q) V

(Q V R) (P /\ R)

T T T T T T T T

T T F T T T F T

T F T T T F T T

T F F F F F F F

F T T T F F F F

F T F T F F F F

F F T T F F F F

F F F F F F F F

t t

Observe that the fifth and eighth columns of this table are identical. That
is, the propositions Pl\ (Qv R) and (Pl\ Q) V (P/\ R) are logically equivalent.
D

EXERCISE SET A.1-B

In Exercises 1-10 use a truth-table to determine whether the given com-
pound proposition is a tautology.

1. P V rv P 6. (P V Q) ==? P


2. P=?P 7. (Pv Q)=?(Pv R)


3. P ==? (P V Q) 8. P ==? (Q ==? P)


4. (P /\ Q) ==? Q 9. (P ==? Q) ==? P


5. rv (P /\ rv P) 10. [(P ==? Q) /\ rv QJ ==? rv P
   In Exercises 11-20 verify the given equivalence using a truth-table.


6. rv (P v Q) = (rv P /\ rv Q)


7. P ==? Q = rv (P /\ rv Q)


8. p v Q = f"V Q ==? p


9. p v Q = f"V p ==? Q


10. rv (P /\ Q) = Q ==? rv P


11. P ==? (Q ==? R) = (P /\ Q) ==? R
    17.P{::}Q = (P=?Q) /\ (Q=?P)


12. rv (P {::} Q) = (P /\ rv Q) v ( Q /\ rv P)


13. rv (P {::} Q) = (P {::} rv Q)


14. PV (QI\ R) := (PV Q) /\ (PV R)