truth tables

For (A AND B)
 If     A      false   true
---------------------
  B    |      A AND B
false | false false
true  | false true
 
**************************

For (P OR Q)
  If    P      false true
---------------------
  Q    |     P OR Q
false | false  true
true  | true true

**************************

What about the case in which P is false and Q is true? In a sense we have no evidence
about the implication as long as P is false. Logicians consider that in this case the
assertion P IMP Q is true. Indeed, the proposition P IMP Q is considered vacuously true
in the case where P is false,...

For (P IMP Q)

 If   P       false   true
---------------------
  Q     |     P IMP Q
false | true false
true  | ? true

**************************

vacuously true or, "The Emporer's New Clothes", white lies, etc.

"Every element of the empty set is also an element of {1,2}" is said to be true,
when in fact there aren't any elements in the empty set.
**********************************************************************************************

For (A AND B) IMP B
 If   A false true
--------------------
  B     |     (A AND B) IMP B
false | true true
true  | true true
 
...the proposition (A AND B) IMP B is true regardless of what A and B stand for.
ie. a tautology.

Taken without permission from from:

-http://www.cs.cornell.edu/Courses/cs3110/2011sp/lectures/lec13-logic/logic.htm#2