This algebra is based on “how to define whether something is true or false”, using symbols and algebraic rules. Abbreviations for certain elements are used, an example of this algebra will be shown, with a problem of Knights and Knaves which will be compared to the math’s results. Boolean Algebra
Boolean algebra is a set with two binary operations: U (union) and N (intersection). It can also be expressed as a logic version using ^ and v. Ex: Set version A U (B N C)= (A U B) N (A U C) Logic version A v (B ^ C) (A v B) ^ (B v C)
John says: We are both knaves. Who is who? Hmmm…. What do you think? Logically do you believe John is telling the truth or not?
Ok! Now that you have in mind who you think is a knight or a knave, Bill being the other person who hasn’t spoken yet, let me tell you a little bit about Boolean Algebra and how it applies to Knights and Knaves. We can use Boolean Algebra to deduce who is who, by using factorization and F.O.I.L.
Let J be true if John is a knight and let B be true if Bill is a knight. Now, either John is a knight and what he said was true, or John is not a knight and what he said was false. Translating that into Boolean algebra, we get: [J ^ (J' ^B')] v [J' ^ (J' ^ B') ' ] = tautology
Simplification Process [J^(J’^B’)] v [J’^(J’^B’)’] False v [J’^(J’^B’)’]; contradiction because J^J’ says that J is a knight and knave The ’ outside makes the sign flip over and the other two ’ are eliminated and it becomes 0 ^ [J ^ B’] v [J’^(J v B)]; By F.O.I.L. 0 v [(J’ ^J) v (J’ ^ B)] Since (J’ ^ J) is a contradiction the answer is (J’ ^ B) Therefore J is a knave and B (who is not mentioned in the problem) is a knight.
Since we’re in the mood…. There’s a fork in the road. One way is to Truths Ville (on the right) and the other one is to Liars Ville (on the left). I want to get to Truths Ville. People from Truths Ville always tells the truth and people from Liars Ville always lie. There is no sign at the fork. I encounter a person (Tom) at the fork and ask him where is Truths Ville? He points to the right. A guy (Eddie) walks out from the right and he said “no Truths Ville is that way”, pointing to the left. Tom says: “ we are both knights.” Who is the knight?
Let’s see what we get! Let’s assume Tom is a knave: [T ^ E] This means: [T ^ (T ^ E)] v [ T ’ ^ (T v E)’] [T ^ E] v [(T ’ ^ (T ’ v E ’)] [T ^ E] v [( T’ ^ (T ‘ v E’)] T ^ E v [( T ’ ^ T ’) v (T ’ ^ E ’)] (T ^ E) v [(T ’ v (T ’ ^ E ’)] (T ^ E) v [(T ’ v T ’ 6 (T ’ v E ’)] Since (T’^ T) is a contradiction that would imply that (T’^ E) is the correct answer. This would mean that Tom is a knave and Eddie is a knight, so the right way to Truths Ville would be the path on the left.
We’re not done yet!!! Boolean Algebra is not that simple, and it takes time to actually let it soak in. The first thing we must do when coming across a certain problem, is organize all the possible conclusions and from there extend our problem and eliminate all false conclusions, and things that contradict each other.
Sources Used: Internet, web pages such as: http://www.math.csusb.edu/notes/sets/boole/boole.htm l http://www.answers.com/topic/logic-puzzle We also received help from Shirley and Anusha. If you are interested in this very complicated but interesting mathematical procedure visit these web pages, and they will be of much help to you.
This project was elaborated by Le Van and Reyna Florentino. With help from SEARCH staff. This topic became interesting to us because of the fact that we did not know that with algebra you can find out something that is intended to be found out by logic. There are many other things that Boolean Algebra is utilized with, such as scientific experiments, and physiological situations. I hope that with our brief presentation we were able to clarify the process of discovering, or revealing “who is who?” Thank You July 21, 2005 4:00 pm