If Then Constraints in DILR: Contrapositive and Chains

DILR

Selection sets are full of sentences like if A goes, then B goes, and if B goes, then C does not. They look simple, and then a question asks who can be in the group if C is present, and the whole thing falls apart on the rough sheet.

The fix is a small amount of logic, applied the same way every time. You need the contrapositive, a way to chain conditions, and a table that translates tricky words like unless and only if. This guide teaches all three with one example you can verify line by line.

Why If Then Sets Are Tricky

If then sentences are natural in language and slippery in logic. People read if A goes then B goes and quietly add that if B goes then A goes. The sentence did not say that. B may go on its own, with or without A.

In a selection set, every condition limits who can be together. A mistake in reading one condition spreads, because you use it to rule out groups, and then your count of valid groups is wrong. The rest of the set then gives answers that look reasonable and are wrong.

So the first job is to read each condition exactly as written, and to turn it into a symbol. A short arrow notation is enough. Write A to B for if A goes then B goes. Write A to not C for if A goes then C does not.

The Contrapositive

Every if then statement has a twin that is true whenever it is true. If A then B is equivalent to if not B then not A. Take the sentence if it rains, the match is cancelled. If the match was not cancelled, it did not rain. The second says the same thing as the first.

This twin is the contrapositive, and it is the most useful tool in the chapter. It lets you reason backward. If you know B is absent, you know A is absent. In a set, that is how an absent person rules out others.

Do not confuse it with the converse, if B then A, or the inverse, if not A then not B. Neither follows from the original. If the match is cancelled, it may have rained or it may not. A cancelled match does not prove rain.

Statement	Equivalent form	Not equivalent
If A then B	If not B then not A	If B then A
A only if B	If A then B	If B then A
A unless B	If not B then A	If B then not A
Neither A nor B	Not A and not B	Only one of them absent

Chain Handling

Conditions often link into chains. If A then B, and if B then C, then if A then C. You can follow the arrows as far as they go. The contrapositive chain runs backward: if not C then not B, and if not B then not A.

Chains grow longer when a condition mentions a negative. If B then not C links with if not C then D, giving if B then D. The rule is to follow arrows from what you know, adding each consequence, until nothing new appears. Then do the same for the contrapositives of every condition.

It helps to draw the chain as a small diagram, with a plain arrow for each condition and a crossed circle for a negation. Drawing it once, instead of rereading the sentences each time, saves minutes in a set with several questions.

A Worked Example

Four people, A, B, C and D, are being considered for a trip. The conditions are these. If A goes, B goes. If B goes, C does not go. If D does not go, C goes. Write them as arrows: A to B, B to not C, and not D to C.

Now add contrapositives. From A to B comes not B to not A. From B to not C comes C to not B. From not D to C comes not C to D. These six statements are the working tool.

Question one: if A goes, who else must go and who cannot? A goes, so B goes. B goes, so C does not. C does not go, so D goes. The group is A, B and D, and C is out. Question two: if C goes, who cannot? C goes, so B does not, from C to not B. B does not go, so A does not, from not B to not A. D is free. The groups are C alone, or C and D.

Question three: how many valid groups exist? Test every possibility against the three rules. The valid groups are D alone, C alone, C and D, B and D, and A, B and D. That is five. The empty group fails, since if D does not go, C must. You can confirm the list by taking each group and checking all three conditions.

Translating Tricky Words

Most errors come from words. A only if B means that A can go only when B goes, which is A to B. It does not mean that B guarantees A. Beginners flip it, and the sentence reads as if B then A, which is the converse.

A unless B means that A goes unless B goes, so if B does not go, A goes. That is not B to A. It does not say what happens when B goes. A may or may not go in that case, unless the set says otherwise. Many students read unless as if B then not A, which adds an unstated rule.

Neither A nor B means both are out. At most one of A and B means they are not both in, which is A to not B. Exactly one of A and B means one is in and one is out. Spend a few minutes on a list of these phrases and their symbols, and the language stops being a source of errors.

If and Only If, and Mixed Conditions

Some sets give if and only if, which is a two way link. A goes exactly when B goes. In that case, treat A and B as one unit that is either in or out together. That simplifies a four person set to three units.

Mixed conditions also appear, with and or or inside the if. If A and B go, then C goes: both A and B must be present to trigger C. Its contrapositive is if C does not go, then A does not go or B does not go, which means at least one of them is absent. If A or B goes, then C goes: either one triggers C, so the contrapositive is if C does not go, then neither goes.

The change from and to or when you take the contrapositive is a source of mistakes. Say the contrapositive aloud once, slowly, for each mixed condition, and write it down. It takes ten seconds and prevents a wrong inference.

Connecting to Other Set Types

If then logic is not limited to selection sets. It appears in binary logic sets, where statements about truth and falsity depend on each other, and in scheduling sets where one event forces another. The same translation and contrapositive habit applies.

It also supports what if questions, which ask for the result of an added condition. Our guide on what if questions without rebuilding the set shows how to apply a new condition to an existing table, and the arrows here make that quick.

For arrangement sets, a different toolkit applies, and our guide on seating arrangements and the diagram lock covers it. Practise both styles, because a section often contains a mix.

A Practice Routine

Start with translation drills. Take twenty sentences with unless, only if and neither nor, and write each as an arrow. Check against a key. This is dull and effective, and a week of it removes most language errors.

Next, take small sets of four or five items and practise the full routine: arrows, contrapositives, chain, and a count of valid groups. Time the first read, and do not let it run beyond a few minutes. Then try binary logic practice to see the same skills in a different setting.

A CAT study planner can schedule these sessions, and you can talk to a CAT mentor if you want feedback on your notation. DILR set types swing between years, so the method matters more than any chapter weight.