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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 \ No newline at end of file