Example

__Example__
Suppose Amelia encounters three women along the road and asks them if the road goes to the capital and whether the bus stop is here.

She receives three different answers:

"The road goes to the capital, and the bus stop is here." "The road does not go to the capital, and the bus stop is here." "The road goes to the capital, and the bus stop is not here."

Amelia is puzzled and asks the women whether they are truthtellers or liars.

This time the three answers are all the same: "Two of us are truthtellers, and one is a liar."

How many of the women are truthtellers? Does the road go to the capital? Is the bus stop at that location?


 * Solution:**

Since the three women all gave the same statement, "Two of us are truthtellers, and one is a liar," they must all be truthtellers, or all must be liars. The statement must be false, so all are liars.

Build the truth table for the other propositions: Let C be the statement, "The road goes to the capital." Let B be the statement: "The bus stop is here."

Let w1, w2, w3 represent the three women.


 * C || B || w1: C^~B || w2: ~C^B || w3: ~C^~B ||
 * T || T || F || F || F ||
 * T || F || T || F || F ||
 * F || T || F || T || F ||
 * F || F || F || F || T ||

You see by the truth table that on no line of the table do exactly two women make a true statement, therefore they are all liars. Only in the first row of this truth table do all women give false statements, therefore, C and B are both true. The road goes to the capital and the bus stop is at that location.

I hope this examples help you.