# Homework Help: Truth Tables

1. Jul 12, 2011

### Shackleford

I'm taking Abstract Algebra right now, and we just briefly covered Logic and Truth Tables. This is my first time in school to learn such things.

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110712_195458.jpg [Broken]

37. I understand.

39. I understand.

41. I don't understand the last column. Is the implication (p ∧ q) ⇒ p true because (p ∧ q) gives you no information on whether p is true? Why?

43. For p ⇒ q, I understand the first two column entries (T,F), is the implication because p being False gives no information on whether p ⇒ q is True? Again, for the last column, is
(p ∧ (p ⇒ q)) True because it being False gives you no new information on q?

If my reasoning is correct, then I can see the following logical consistencies here. Also, I notice for the implications ⇒, there is an additional column asking for its truth value. But, there is no such thing for the iff ⇔ statements. Why?

Last edited by a moderator: May 5, 2017
2. Jul 12, 2011

### lanedance

41 is called vacuously true, see
http://en.wikipedia.org/wiki/Vacuous_truth

basically in a statement
$$P \implies Q$$

if P is false, then the statement is vacuously true, as the rest of the statement no longer needs to be evaluated

consider in the form
if P then Q
when P is false the statement has no more information. In an analogy with programming, there is no elseif or else so nothing else to evaluate making it vacuously true

3. Jul 12, 2011

### SammyS

Staff Emeritus
41. is true because the only way for $A\implies B$ to be false is for A to be true and B to be false.

(p AND q) is always false when p is false.

4. Jul 12, 2011

### SammyS

Staff Emeritus
I would word this as: When p is false , then p ⇒ q gives no information as to whether q is true or false.

5. Jul 13, 2011

### Staff: Mentor

Another way to think of this (#41) is this: If p and q are true, then obviously p is true. There's a related statement -
(p ∧ q) ⇒ q

6. Jul 14, 2011

### Shackleford

Okay. I got it. Thanks for the help, guys.