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Teach: which way does the rule go?

Every rule goes one way. Find which way before you read the options.

If-then rules take you from one thing to another. They don't take you back. You'll learn to say which way a rule goes, then only move that way.

'If it rains, the match is off' goes one way. Rain means no match. But no match doesn't mean it rained. The match could be off for another reason.

'Only if' works differently. 'You get a badge only if you finish the log' means a badge needs a finished log. So badge, then log. Finishing the log doesn't promise a badge.

There's one safe flip. If the second part didn't happen, the first part didn't either. No finished log means no badge. That's always true.

The trap is reading the rule backwards. 'Omar finished the log, so he has a badge' turns the rule around. The rule never says that.

So write the rule as 'this, so that' before you read the options. Then only move forward, or use the safe flip.

If-then rule

a rule that takes you from one thing to another, one way only.It tells you which way you can move.

Only if

sets a must-have. 'A only if B' means A, so B.It's the rule people most often read backwards.

Safe flip

if the second part didn't happen, the first part didn't either.It's the only way to use a rule from the end.

How to solve an if-then question

What's askedA badge is given only if the log is done. Leila has a badge. Omar did his log. What must be true?
The facts'Only if' means badge, so log done. It doesn't go the other way.
Work it outLeila has a badge, so her log is done. Omar's log doesn't tell us about a badge.
The trap'Omar has a badge' reads the rule backwards.
The answerLeila completed the service log.

The moveRead the question. Use the method. Then check the tempting option.

You can say which way a rule goes.
You can use the safe flip.
You can spot an option that reads the rule backwards.

show

Show: a worked example

Watch how to solve it, one step at a time.

Read the problem, then follow the steps. You'll see the answer, the trap, and what to do next time.

Full problem: one 'only if' rule, two students

Question: The badge rule uses 'only if'. Leila has a badge. Omar finished the service log. Which must be true?

The service log

A school badge is given only if a student finishes the service log.
Leila has a school badge.
Omar finished the service log.

Which of these must be true?

A.Omar has a school badge.

B.Leila finished the service log.

C.Every student who finishes the service log gets a badge.

D.Leila finished the service log before any other student.

Step 1 — Read the question
Work out what kind of answer you need.
The question asks what must be true. There's one rule, and it uses 'only if'. So first I'll work out which way the rule goes.

'A badge only if the log is finished' means you can't have a badge without a finished log. So badge, then log done.
- So first I'll work out which way the rule goes.Do this before you look at Leila or Omar. Then the options can't push you the wrong way.
- So badge, then log done.'Only if' points to the must-have. The must-have comes second.
Step 2 — Work it out
Follow the steps from the facts to the answer.
Put each student on the rule.
- Rule: badge, so log done.
- Leila has a badge. So Leila finished the log.
- Omar finished the log. The rule doesn't go from log to badge. So we don't know if Omar has a badge.
Answer: B.
- So Leila finished the log.Leila starts at the front of the rule, so she can move forward.
- So we don't know if Omar has a badge.Omar is at the end of the rule. You can't move back from there.
Step 3 — Spot the trap
See why a tempting answer is wrong.
Common wrong answer: A, 'Omar has a school badge.'

Why it tempts you: Omar did the thing the rule talks about. It feels like he's earned the badge.

Why it's wrong: the log is a must-have, not a promise. Some students might finish the log and still not get a badge. The rule only says no log means no badge.

Fix: write the rule as 'badge, so log done'. Then check if the option moves backwards.
- the log is a must-have, not a promiseA must-have only rules people out. It never rules anyone in.
- Then check if the option moves backwards.A goes from log to badge. That's the wrong way.
Step 4 — The answer
Say the answer and the shortest proof.
B. The rule says a badge needs a finished log. Leila has a badge, so she finished the log.

A reads the rule backwards. C says the same thing as a rule for everyone. D adds 'before any other student', which no fact gives.
- Leila has a badge, so she finished the log.One rule, one forward step. That's the whole proof.
Step 5 — The method
What to remember for questions like this.
Write every rule as 'this, so that'. Then you can only do two things:
- Move forward: badge, so log done.
- Use the safe flip: no log done, so no badge.
Anything else is a trap. Moving backwards (log done, so badge) isn't allowed. Neither is flipping both parts (no badge, so no log).
- Move forward: badge, so log done.Use this when you know the first part.
- Use the safe flip: no log done, so no badge.Use this when you know the second part didn't happen.

Step 6 — Next time

What to do on your next question.

In this question	Next time	Why it helps
Wrote 'only if' as 'badge, so log done'.	Write the rule as 'this, so that' first.	You can see which way it goes.
Moved forward from Leila.	Only move forward, or use the safe flip.	Those are the only safe moves.
Rejected A, which went backwards.	Ask: does this option move the wrong way?	It catches the most common trap.

Write the rule as 'this, so that' first.Do it before reading the options.

compare

Compare: a guess, or a proof?

Picking what sounds right, or following the facts.

The weak answer picks what sounds sensible. The strong answer follows the facts and stops where they stop.

The whole response

Sounds right

Jack returned all his books, so he must have a library pass.

Follows the facts

A pass needs all books returned. Mei has a pass, so she returned all her books. Jack's returned books don't prove he has a pass.

Point-by-point breakdown3

First moveWeaker: Looks for something that sounds sensible.Stronger: Writes the rule as 'pass, so books returned'.It keeps the answer tied to the facts.
ProofWeaker: Gives an answer with no link to the facts.Stronger: Shows each step from the facts.Every part of the answer needs a fact behind it.
The trapWeaker: Reads the rule backwards.Stronger: Says why the trap fails.A rule only goes one way.

guide

Guide: try one together

One small step with help before you go solo.

Look back at the worked problem. Now try one like it.

A student may enter the lab only if a teacher has said yes. Mia entered the lab. Jack has a teacher's yes, but he hasn't entered. What must be true? Pick one. Then say which way the rule goes, and what happens if there's no teacher's yes.

Want feedback on your own answer? Get started to practise with instant marking.

Say which way the rule goes.
Say what follows for Mia and for Jack.
Use the safe flip for no teacher's yes.

Your turn

A club badge is given only if a member finishes training. Leo has a badge. What must be true?

A. Leo trained before anyone else
B. Leo runs the club
C. Leo finished training
D. Everyone who trained has a badge

If a gate alarm goes off, the gate locks. The gate did not lock. What must be true?

A. The gate was never opened
B. The alarm is broken
C. The alarm went off
D. The alarm did not go off

If-Then and Only-If Rules