for
Section 4: Tautological Implications and Tautological Equivalences
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11. p = "some cows are chickens;" q = "some chickens lay eggs." Then pq is true, whereas p is false. Thus, (p
q)
q is false.
13. p = "all swans are white;" q = "some swans are white." Then pq is true (since the statement p is false; not all swans are in fact white). On the other hand, q
p says that, if some swans ar white, then all swans are white. But some swans are white, so that q is true; whereas p is false. Thus, q
p is false. Therefore, (p
q)
(q
p) is false.
15. Use the same example as in (13).
17. (ht)
h Tautology
19. ~(rv)
((~r)
(~v)) Not a tautology
21. (ur)
(~r
~u) Tautology
23. (ur)
(~r
~u) Tautology
25. (gp)
(g
~p) Not a tautology
27. ((th)
~t)
h Tautology
29. ((gs)
(g
j))
(s
j) Not a tautology
31. ((gs)
~g)
~s Not a tautology
![]() | 2. Logical Equivalence, Tautologies, ... | ![]() | 3. The Conditional and the Biconditional | ![]() | 4. Tautological Implications and Tautological Equivalences | ![]() | Main Logic Page | ![]() | "Real World" Page |
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![]() | Stefan Waner (matszw@hofstra.edu) | ![]() | Steven R. Costenoble (matsrc@hofstra.edu) |