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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)
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.
5. No.
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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