More guilt
The following statements are all true:
If A is not guilty, then B and C are both guilty.
Either A is not guilty or B is guilty.
Either B is not guilty or C is not guilty.
Who is guilty?
If A is not guilty, then B and C are both guilty.
Either A is not guilty or B is guilty.
Either B is not guilty or C is not guilty.
Who is guilty?
Labels: logic
posted by Chris at
6:02 PM
4 Comments:
A cannot be not guilty (NG), then the last statement would be false for B and C cannot be both (guilty) G while either is NG.
therefore A must be G, so from the second statement we know that A cannot be NG, therefore B is G
Because B is G, then B cannot be NG so C is NG
Answer - A is guilty, B is guilty, C is not guilty
Yes?
Hi Jayson - "yes" indeed, well done. I struggle with these types of statements.
A formal way of doing this is to note that, if A then X is true (1) is equivalent to /A + X = 1, where / is the negation operator. This can also be written A -> X (implies).
Also AB => A AND B, A/A = 0, AA = A.
The three statements can then be written (using A, B and C to denote A is guilty etc.):
A + BC = 1, /A+B = 1, /B + /C = 1.
AND the LHS expressions together and equate to 1, then expanding yields:
(A+BC)(/A+B)(/B+/C) = 1
(A+BC)(/A/B + /A/C + B/B + B/C) = 1
A/A/B+A/A/C+AB/C + BC/A/B+BC/A/C+BCB/C) = 1
=> AB/C = 1
so A and B are guilty and C is not guilty.
I've just started to learn this stuff, hence the post.
I agree. Considering my programming background, I'm surprised that I didn't use ! or ~ up front. Thanks for nudge.
In the following I use 1 and 0 to represent true and false (and not the cardinal numbers 1 and 0). I am partly rewriting it as I'm trying to understand the "algebra of statements" better and to cover (any) mistakes I've made.
In the algebra of statements, if 'if A is true then X is true' is true, is abbreviated to 'if X then Y', and if the statement is false to '~(if X then Y)'. The former statement is often written 'X->Y' (read as X implies Y) and is equivalent to '~A + X', where ~ is the logical negation operator. I can't bring myself to go the whole hog on this (yet), so I'll compromise with '~X+Y = 1'. To me this is a whole new way of looking at logic, it's my new toy and I'm playing with it.
Back to the problem. The three statements can then be written (using A, B and C to denote A is guilty etc.):
A + BC = 1, ~A + B = 1, ~B + ~C = 1.
AND the LHS expressions together and equate to 1
(A+BC)(~A+B)(~B+~C) = 1
(A+BC)(~A~B + ~A~C + B~B + B~C) = 1
A~A~B+A~A~C+AB~C + BC~A~B+BC~A~C+BCB~C) = 1
=> AB~C = 1
So A and B are guilty and C is not guilty. Note that if you mentally drop the '= 1' bits, it still makes sense.
Post a Comment
Links to this post:
Create a Link
<< Home