Trolls either always lie or always tell the truth.
Question
You are given that:
Trolls either always lie or always tell the truth.
Two trolls claim the following:
Troll 1: If we are brothers, then we are both honest.
Troll 2: We are brothers or we are both honest.
So, Which statement is true?
And Draw the truth table.
Also, Describe how you would find the solution. You must refer to the truth table. And Give the solution.
Explanation
Here we have given that Trolls either always lie or always tell the truth.
To solve the problem:
So State the premises in logical terms.
Let, P: we are brothers, and
Q: We are both honest.
Also. For Troll 1: Q→P
For Troll 2: P OR Q
The truth table is as follow
| P | Q | P OR Q | P AND Q | P→Q | Q→P |
| T | T | T | T | T | T |
| T | F | T | F | F | T |
| F | T | T | F | T | F |
| F | F | F | F | T | T |
So, To find the solution for this, we match all the cases which are present. And conditions for the statements and troll. As given in the question that the troll either lies or tells the truth, so if we find anyone condition which does not satisfy the trolls. And If one is not the answer then the second one is the answer.
In the case of Troll 1, when Q is false and P is true. And the result is true.
The statement says that “If we are brothers, we are honest.”
So, this says that they are not honest in one case, i.e., at that time they are not brothers. It is not possible that they are dishonest if they are brothers.
So Q must be true whenever P is true.
Also, we can conclude that Troll 2 is true and Troll 1 is incorrect as in one case it gives inappropriate results.
Also read, Simplify this expression using Boolean algebra rules.