What is Binary Logic?
In CAT, Binary logical reasoning concept is used to frame the questions on Truth teller, Truth-liar and alternator.
- Binary logic consists of a set of rules based on operations of mathematical binary operations of NOT, AND, and OR.
- There are only two logical propositions either True or False.
1. TruthTeller: This person will always speak the truth. All the statements made by this person are true.
2. Liar: This person will always tell a lie. All statements made by this person are false.
3. Alternators: This person always alternates between the truth and the lie.
- If first statement of this person is true, then second will be false, third will be true and so on.
- Similarly, if first statement made by this person is false, then second will be true, third will be false and so on.
- There is no particular number of true statements or false statements made by this person but order is always TRUE-FALSE-TRUE- or FALSE-TRUE-FALSE.
Assumptions in Binary Logic
- Binary logic questions are all about making some assumptions (kind of assumptions are described later).
- These assumptions may give rise to some contradictions which are the indicators that our assumptions are wrong.
- If for any assumption we do not get even a single contradiction, then that is the solution for the given binary logic problem.
Kinds of Problems Asked in Binary Logic
- The binary logic problems, as told earlier, will contain people who speak binary statements. The very general trend is that each of them will make 3 statements.
- The number of people varies from 3 to 5. Generally, only 3 person problem is asked in the CAT.
- Now, the question will specifically mention how many of them are truth-teller, liar or alternator. It is not always that we have a person of each category.
- There are cases where all 3 can be alternator or 1 truth-teller, 2 alternator, etc. Also, when the number of persons is increased to 4 or 5 the question becomes even more complicated giving rise to more iterations.
Step Wise Solved Examples
Example 1: Three boys- Aman, Bagheer, and Chiru replied to the question, “Who among you is a Doctor” in the following manner:
We know exactly one of these boys is a Doctor, one is a Painter and one is an Athlete.
Further, one always speaks the truth, one always lies, and one alternates between the truth and the lie.
Steps to Solve
Step 1: With the help of some statements made by these people, we might be able to identify them without any assumption. Here is a list of 4 statements that one must always look out for to make the task easier. If any one of these is made by any person, then we can categorize them as explained below.
1. I am a liar
- Consider if a truth-teller says, “I am a liar”, which is a lie as a truth-teller can only say, “I am a truth-teller”. Hence, we can conclude that the person who said “I am a liar” is not the truth-teller.
- Similarly, if a liar says that he/she is a liar then that statement will be true but the liar will always speak the lie. None of the statements made by him/her can be true. We can conclude this statement cannot be made by a liar.
- The only category of person who can speak this statement can be the alternator. He can alternate between the truth and the lie. Since he/she is not a liar but he/she can still make a false statement, alternator is the only category of people who can make the statement, “I am a liar”. The statement in itself will be a lie.
- This gives us another hint that the statement proceeding and the statement preceding this statement will always be true.
2. I am not a truth-teller
- Similar to the explanation above, a truth-teller can never make this statement because if he/she makes this statement then it will be a lie which contradicts the fact that a truth-teller always speaks the truth.
- If the liar makes the above statement, then it will be the truth for him which again contradicts the fact that a liar will always lie.
- Hence, a liar cannot make this statement.
- The alternator can say, “I am not a truth-teller”, as he can say either the truth or the lie. This statement will be a true statement for him, which gives us another hint that the statements preceding and proceeding this statement are the lie.
3. I am an alternator
- A truth-teller cannot make this statement as this statement will be a lie for him which conflicts the fact of the truth-teller.
- A liar can make this statement as this statement will be a lie for him/her.
- An alternator can also make this statement and this will be the truth for the alternator.
- We can conclude that this statement can be made by the liar or the alternator.
4. I am not an alternator
- Similar to the above statement, a liar will not make this statement.
- A truth-teller can make this statement.
- An alternator can make this statement and this time it will be the lie for him/her.
Summary to Explain Example 1
- In our example question, Bagheer makes the statement, “I am a liar”, which implies that Bagheer is the alternator. Also, Aman said that he is an alternator, which implies that he can be either a liar or an alternator. Now, since Bagheer is the alternator, Aman is definitely the liar.
- This leaves us with Chiru as the truth-teller. Since Bagheer is the alternator, so the statements proceeding/preceding the statement, “I am a liar” are true, we can conclude that Aman is not a Doctor.
- Also as Chiru is a truth-teller, according to his statements, Bagheer is not a painter and he himself is not a painter, thus Aman is the painter. Now, Bagheer’s first statement is a lie (since third statememt is also a lie, so the order for Bagheer’s statements will be 1^{st}-false,2^{nd}-true,3^{rd}-false), this means Chiru is not an athlete, which leaves us with the only option for Chiru as Doctor and Bagheer will then be an athlete.
- To summarize our findings:
So the answer to the question, ‘Who is a Doctor’, is Chiru.
Other statements such as, “I am a truth-teller”, can be made by all the three categories of person and so will not be of much help to us.
Step 2
- In some cases, where none of the four statements as mentioned in step 1 was made by anyone, we will use the assumption-iteration method.
- In this method we will assume the first person as the truth-teller and based on his statements we will try to find conflicts or contradictions that may arise due to the statements made by others.
- We will also use the assumptions-iteration method when statement 3 or 4 mentioned in step 1 is made by anyone as these statements will leave us with two options.
Example 2: Utkarsh, Ravi, and Shivam made the following statements regarding the type of vehicle they own. Each one of them belongs to exactly one category of truth-teller, liar or alternator. Only one among them is a truth-teller. Further, we know each of them own a different vehicle and each of them own exactly one among car, cycle and bike.
Solution:
- Now, Utkarsh says that he is not an alternator and as per step 1 we know that he can be a truth-teller or an alternator.
- To proceed with the approach, we will first assume that Utkarsh is the truth-teller and all the statements are true. We know exactly one among them is a truth –teller and we already assumed Utkarsh as the truth-teller.
- The question does not specifically mention the exact number of each category. From Utkarsh’s statements, we know neither Shivam owns a car nor Ravi. Reading Ravi’s statements, since his last statement is true so he should be an alternator (he can’t be the truth-teller as there is only one truth-teller which we already assumed as Utkarsh).
- This implies his second statement should be false but as per the statement made by Ravi, the second statement is also true.
- This is a contradiction to our assumption which means that our assumption is wrong.
We have to go for the second interation knowing Utkarsh is the alternator. - As he is not the truth-teller, he has to be an alternator. His second statement is false which automatically makes his first and second statements true.
- We can deduce that Utkarsh owns a car as none of Ravi and Shivam owns a car as per Utkarsh’s statements. Shivam’s first statement is a lie so he can be a liar or an alternator. Since there is one truth-teller and the only option for the truth-teller is Ravi thus we can say Ravi is the truth-teller.
- This makes Shivam’s second statement true and so we know that Shivam is an alternator. Shivam does not own a cycle makes Ravi own a cycle. Shivam owns a Bike.
Summary
- We conclude this post with this example where there are two alternators and one truth-teller.
- The question specifically mentions the number of truth-teller in order to avoid any ambiguity.
- Always look for Step 1 statements in order to make the task simpler and time-saving and then go for the assumption-iteration method.