Decisions and Boolean reasoning

By the end you can:

• Translate a written rule into a correct relational and Boolean condition.
• Evaluate AND, OR and NOT, and find an input where AND and OR give different results.
• Trace which branch runs for a given input, including at a boundary value.
• Explain why two independent IF statements can both run while IF ELSE runs exactly one block.
• Trace a nested conditional by following only the branches actually taken.

Boolean value

A value that is either true or false.

Relational operator

An operator that compares two values and produces a Boolean, such as ≥ or ≠.

Logical operator

AND, OR or NOT, which combine or flip Boolean values.

Conditional statement

A statement that runs a block only when a condition is true (IF), optionally with an alternative (ELSE).

Boundary value

An input exactly at, or just either side of, the point where a rule changes its answer.

Nested conditional

A conditional statement placed inside a branch of another conditional.

A Boolean value is either true or false. Relational operators compare two values and produce a Boolean: =, ≠, >, <, ≥ and ≤ in AP pseudocode (==, !=, >, <, >=, <= in Python).

Our running example is a club eligibility checker. The rule from the club is: a student may join if they are at least 14 and the club has more than 0 places. “At least 14” means age ≥ 14, so 14 itself is allowed. “More than 0” means places > 0, so 0 is not.

a AND b is true only when both are true. a OR b is true when at least one is true (including when both are). NOT a flips a single value.

AND and OR agree on the first and last rows and disagree on the middle two. Those middle rows are exactly the inputs that reveal which operator a program uses.

NOT (old AND room) means “not both”. It is the same as (NOT old) OR (NOT room). It is not the same as (NOT old) AND (NOT room), which means “neither”.

1IF ((age ≥ 14) AND (places > 0))2{3  DISPLAY("You can join")4}5ELSE6{7  DISPLAY("Not eligible")8}

The condition is evaluated once. If it is true, only the first block runs; otherwise only the ELSE block runs. Exactly one of the two messages is displayed.

Two separate IF statements are different: each condition is checked on its own, so both blocks can run, or neither.

Worked example

A student wrote this checker. Predict what it displays for a 14-year-old when the club has 1 place.

1IF ((age > 14) OR (places > 0))2{3  DISPLAY("You can join")4}5ELSE6{7  DISPLAY("Not eligible")8}

Fix 1: replace OR with AND. Re-test (14, 1): now age > 14 is false, so the fixed-OR version wrongly says not eligible. Fix 2: replace > with ≥ for age. The repaired condition is (age ≥ 14) AND (places > 0), and it gives the rule’s answer for all four tests.

A conditional can sit inside another. The inner condition is only checked when the outer branch containing it runs.

1IF (member)2{3  IF (places > 0)4  {5    DISPLAY("Booked")6  }7  ELSE8  {9    DISPLAY("Waitlist")10  }11}12ELSE13{14  DISPLAY("Join the club first")15}

For a non-member, the places check never happens and “Join the club first” is displayed, even if places is 10. To trace a nested conditional, follow the outer condition first, then only the inner block on the path you took.

• AND and OR are interchangeable

  The belief: Both join two conditions, so either works.

  What is actually true: They differ exactly when one part is true and the other false. Test those inputs to see which one the program uses.

  Check yourself: Can you give an input where your condition is true with OR but false with AND?

• Only one IF can run

  The belief: Two IF statements in a row work like IF ELSE.

  What is actually true: Independent IF statements each check their own condition, so both can run. Only IF ELSE guarantees exactly one block.

  Check yourself: Pick an input that makes both conditions true. What is displayed?

• One passing test proves the condition

  The belief: It worked for my test, so the checker is correct.

  What is actually true: A wrong condition often passes some tests. Test on and either side of each boundary and where the parts disagree.

  Check yourself: Which of your tests would fail if ≥ were changed to >?

1. Give an input (age, places) for which (age ≥ 14) OR (places > 0) and (age ≥ 14) AND (places > 0) give different results.
2. Write a condition for “at most 30 students and at least 1 teacher”, using students and teachers.

Practice only: answers are checked in your browser and nothing is saved or sent to your teacher. Each explanation links back to the section it practises.

Lesson practice

Apply it somewhere new