DILR "Which Is NOT True" Questions: One Counterexample

DILR

You have solved the set. The table is full, the arrangement is fixed, and then the question reads: which of the following is NOT true? Many students re-check all four options against the whole set and lose three minutes. There is a faster way, and it rests on one idea: a statement that is not always true can be killed by a single valid case.

This guide shows how to read the wording of such questions, how to use one counterexample to test an option, and how to choose the order in which to test. A small worked set is included, and every claim about it was checked by listing all valid arrangements.

Two Meanings of NOT True

The phrase is used in two ways, and the difference changes the work. In one meaning, the question asks for the statement that is false in every valid case, a statement that must be false. In the other, it asks for the statement that is not guaranteed, one that can be false in at least one valid case. Questions often say it more precisely, using phrases such as cannot be true, must be false, or need not be true.

The wording of the options helps. If three options are things that are always true, and one option is something that sometimes fails, the question is about the second meaning. If three options are possible and one is impossible, the question is about the first. Reading the options before you decide is a small step that avoids a wrong plan.

When the wording is truly ambiguous, the options usually still make one meaning workable. Choose the meaning under which exactly one option qualifies. If two options qualify under one reading and only one under the other, the question setter meant the second.

The Counterexample Idea

A statement is always true only if it holds in every valid arrangement. So one valid arrangement where it fails is enough to show that it is not always true. You do not need to list all valid arrangements. You need one.

This turns a heavy check into a light one. Instead of proving that an option holds everywhere, which needs a full case analysis, you search for one case where it does not. If you find it, you have your answer for a need not be true question. If after a fair attempt you cannot, the option is probably always true, and you move to the next.

The idea has a limit. For a must be false question, a single case does not help, because you must show the option fails everywhere. There you test by trying to make the option true. If you find a valid case where the option is true, it is not the answer. The one option that cannot be made true is the answer.

A Small Worked Set

Five runners P, Q, R, S and T finish a race with no ties. P finishes before Q. S finishes before R. T finishes either first or last. Which of the following is NOT necessarily true? Option A: Q does not finish first. Option B: R does not finish first. Option C: P finishes before R. Option D: S does not finish last.

Listing all valid orders gives twelve. In all twelve, Q is never first because P must be ahead of Q. R is never first because S is ahead of R. S is never last because R is behind S. P is never last because Q is behind P. So the options A, B and D hold in every case.

Option C is the one that can fail. In the order S, R, P, Q, T, every condition is met: P is before Q, S is before R, and T is last. But R finishes before P. One valid order kills option C, and it is the answer. Without listing all twelve, you could find this order by placing S and R at the front, which are the two runners with fewer constraints on that side.

Option	Statement	Result across the 12 valid orders
A	Q is not first	True in all 12
B	R is not first	True in all 12
C	P finishes before R	True in 10, false in 2
D	S is not last	True in all 12

How to Build the Counterexample

To break an option, set up the case that opposes it. If the option says P finishes before R, build an arrangement with R before P and see whether the rest of the conditions can hold. You are not solving the whole set again. You are fixing one fact and checking whether the set still allows it.

Start by placing the fact you want, then place the most constrained items, then fill the rest. In the race, the fact is R before P, and the most constrained item is T, which must be first or last. Put T at the end, then place S before R, R before P, and P before Q. The chain S, R, P, Q fits in the first four places.

If the build runs into a conflict, the conflict is information. It tells you that the fact cannot hold, and the option is true in all valid cases. Note the conflict, mark the option as safe, and move on.

Choosing Which Option to Test First

Test options in order of how likely they are to break. Options that name a specific position, such as X is third, are often easier to break than general ones, since a single case with X elsewhere kills them. Options that compare two items, such as X is before Y, are broken by one case where the order is reversed.

Options that say never or always are strong claims, and strong claims break easily. Options with weaker wording, such as is not first, hold in more cases. If one option says P is never last and another says Q can be in the second place, start with the first, because a never claim needs only one case.

If you have drawn a few valid cases while solving the set, test options against those cases first. You may find the answer without any new work. This is the main reason to keep your cases written down and not erased.

When the Set Is Larger

In a big set, such as a grid with several attributes, you may not be able to list all valid cases, and the counterexample method becomes the main tool. Fix the fact you want, then propagate the consequences using the conditions. If the propagation ends in a contradiction, the fact is impossible. If it ends with a complete valid grid, you have your case.

The risk is that you will stop too early and call a case valid before you have checked every condition. Before you accept a counterexample, run down the list of conditions one by one and tick each. A counterexample that violates one condition is worth nothing.

Our guide on what if questions without rebuilding the set shows a similar skill: changing one fact and tracing the effect without a fresh start. The same habits help here, and the two question types often appear in the same set.

Linking It to Set Selection

Not true questions can be quick or slow depending on how well you have recorded the set. A set where you ended with a clean table is the one where these questions are cheap. A set where you ended with a cluttered sketch and many open possibilities is one where they are expensive.

When choosing sets in the exam, think about this. If a set has several questions of this kind, the value of a clean setup is higher. Spend the extra half minute at the start to tidy your notation, because it pays back in each question. Our piece on the four step diagram lock shows how to build a diagram that supports this kind of testing.

Do not decide in advance how many questions to attempt in a set. Attempt the ones you can settle with confidence, and park the others. A wrong answer costs 1 mark on an MCQ, and a blank costs nothing.

To fit this into your plan, use the CAT study planner, and if you want a review of how you handle sets, you can talk to a CAT mentor.