7. Show that A <=> B is equivalent to (not A) <=> (not B).
8. Construct truth tables to illustrate the following:
(a) A <=> B.
(b) A => (B and C).
9. Use truth tables to prove that the following are equivalent: A => (B and C) and (A => B) and (A => C).
10. Verify the equivalence in question 9 by means of a logical argument.
Here are a whole bunch of truth tables for 7-9... I'm going to take a break before attempting #10.
ReplyDelete7.
Show that Φ ⇔ Ψ is equivalent to (¬Φ) ⇔ (¬Ψ).
Φ | Ψ | Φ ⇒ Ψ | Ψ ⇒ Φ | Φ ⇔ Ψ
T | T | T | T | T
T | F | F | T | F
F | T | T | F | F
F | F | T | T | T
|
¬Φ | ¬Ψ | ¬Φ ⇒ ¬Ψ | ¬Ψ ⇒ ¬Φ | (¬Φ) ⇔ (¬Ψ)
F | F | T | T | T
F | T | T | F | F
T | F | F | T | F
T | T | T | T | T
8.
a. Φ ⇔ Ψ
Φ | Ψ | Φ ⇒ Ψ | Ψ ⇒ Φ | Φ ⇔ Ψ
T | T | T | T | T
T | F | F | T | F
F | T | T | F | F
F | F | T | T | T
b.
Φ ⇒ (Ψ ⋁ θ)
Φ | Ψ | θ | (Ψ ⋁ θ) | Φ ⇒ (Ψ ⋁ θ)
T | T | T | T | T
T | T | F | T | T
T | F | T | T | T
T | F | F | F | F
F | T | T | T | T
F | T | F | T | T
F | F | T | T | T
F | F | F | F | T
9.
Φ ⇒ (Ψ ⋁ θ)
Φ | Ψ | θ | (Ψ ⋀ θ) | Φ ⇒ (Ψ ⋀ θ)
T | T | T | T | T |
T | T | F | F | F
T | F | T | F | F
T | F | F | F | F
F | T | T | T | T
F | T | F | F | T
F | F | T | F | T
F | F | F | F | T
(Φ ⇒ Ψ) ⋀ (Φ ⇒ θ)
Φ | Ψ | θ | (Φ ⇒ Ψ) | (Φ ⇒ θ) | (Φ ⇒ Ψ) ⋀ (Φ ⇒ θ)
T | T | T | T | T | T
T | T | F | T | F | F
T | F | T | F | T | F
T | F | F | F | F | F
F | T | T | T | T | T
F | T | F | T | T | T
F | F | T | T | T | T
F | F | F | T | T | T
Same answers.
ReplyDeleteAlso same.
ReplyDeleteI would think there might be a program on the web somewhere that uses a Java script or something to generate truth tables. I understand the utility of doing them by hand, but after awhile it becomes a pretty time-consuming task. One small error can be an even greater time-suck. I suppose that's the advantage of having a discussion group, but still I wonder if/where a piece of software exists online for doing these tables, especially when there are three variables.
Well, to answer my own question, here's one: http://www.brian-borowski.com/Software/Truth/ that appears quite well developed (Requires Java 1.6 or later).
I guess we all punted on the logical argument for equivalence of #9, which is Q10.
ReplyDelete