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1)
1. Valid (it is always true)
2. Neither (this is occasionally true)
3. Neither
A⇒B (C) ~A⇒~B (D) C⇒D
T T T
F T T
T F F
T T T
4. Valid (A v ~A is always true)
5. Neither
A B C A⇒B (D) B⇒C (E) D⇒E
T T T T T T
T T F T F F
T F T F T T
T F F F T T
F T T T T T
F T F T F F
F F T T T T
F F F T T T
6. Valid
(A⇒B)⇒((~A|~C)⇒B)
A B C A⇒B (D) ((~A|~C)⇒B) (E) D⇒E
T T T T T T
T T F T T T
T F T F F T
T F F F T T
F T T T T T
F T F T T T
F F T T T T
F F F T T T
7. Valid (at least one is always true)
A B A⇒B
T T T
T F F
F T T
F F T
8. Neither
A B A^B (C) C∨~B
T T T T
T F F T
F T F F
F F F T
9. Valid
A B C (P⇒Q)∧(Q⇒R) (D) D⇒(A⇒C)
T T T T T
T T F T T
T F T F T
T F F T T
F T T F T
F T F F T
F F T F T
F F F T T
10. Valid (this truth table is really long but it turns out to be a tautology)
2) (A^B)⇒C == (~A∨~B)⇒C
1. They are both true iff all 3 are true so no?
2. Yes. This is the statement.
3. No. AND and OR are not interchangable.
4. No.
"If the night is quiet, then both the dog sleeps and the house is warm"
5. No.
"The dog does not sleep or the night is not quiet or the house is warm"
3)
1. ~~(~C∨~D)∨P
-> ~C∨~D∨P
2. ~J∨(Wi^We)
-> (~J∨Wi) [a]^(~J∨We) [b]
3. ~Wi∨D
4. ~We∨C
5. J
--REFUTATION--
6. ~P
7. (1,6) ~C∨~D
8. (7,3) ~Wi∨~C
9. (8,4) ~Wi∨~We
10. (9,2a) ~J∨~We
11. (10,5) ~We
12. (11,2b) ~J
13. (5,12) NIL
4)
1. A∨B∨C
2. ~A∨B
3. ~B [a]^ ~D [b]
4. ~C∨D
--REFUTATION--
5. (1,4) A∨B∨D
6. (3,5) A∨D
7. (6,4) A
8. (7,2) B
9. (8,3) NIL
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