Applying logical operators in truth tables

What you'll learn

• How the logical operators AND, OR and NOT work.
• How to build truth tables with one, two or three inputs.
• How to add intermediate columns so complex expressions become manageable.
• How to use a truth table to solve a simple problem or decision rule.

The basic idea: logic uses True and False

In Computer Science, many decisions are based on whether something is True or False.

For truth tables, we usually represent:

• True as 1
• False as 0

For example, a variable called DoorOpen could be:

• 1 if the door is open
• 0 if the door is not open

Inputs, outputs and logical operators

An input is a Boolean value that goes into a logic expression or logic gate.

An output is the Boolean result that comes out.

A logical operator is a rule that combines or changes Boolean values. For this topic, you need to apply:

• NOT
• AND
• OR

This reference diagram summarises the three operators you need for Edexcel GCSE Computer Science.

NOT: reversing a value

NOT takes one input and reverses it.

So:

• NOT 0 becomes 1
• NOT 1 becomes 0

AND: both must be true

AND takes two inputs. The output is 1 only if both inputs are 1.

A common real-life version is: “You may enter if you have a ticket AND your ticket is valid.” Both conditions must be true.

OR: at least one must be true

OR takes two inputs. The output is 1 if at least one input is 1.

What a truth table does

A truth table lists every possible combination of input values, then shows the output for each combination.

If there are $n$ inputs, the number of rows is $2^n$.

So:

• 1 input gives 2 rows
• 2 inputs gives 4 rows
• 3 inputs gives 8 rows

For Edexcel GCSE 1CP2, you need to handle truth tables with up to three inputs.

Listing input combinations systematically

For two inputs, the usual order is:

For three inputs, count from 000 to 111:

Building a truth table for a combined expression

More difficult questions often combine operators, such as:

(A AND B) OR C

Do not try to do the whole expression in your head. Instead, add an intermediate column for part of the expression.

For (A AND B) OR C, a useful intermediate column is:

A AND B

Then the final output column is:

(A AND B) OR C

Using NOT inside a larger expression

NOT is often used to mean a condition is not happening.

For example:

NOT Locked

This means the output is 1 when Locked is 0.

In a larger expression, calculate the NOT part first by flipping that column.

Solving problems using truth tables

A problem may describe a real system in words. Your job is to turn the words into a logic expression, then use a truth table.

Example conditions:

• CardValid is 1 if a user has a valid card.
• PINCorrect is 1 if the user entered the correct PIN.
• AccountLocked is 1 if the account is locked.

A cash machine might allow access only if:

CardValid AND PINCorrect AND NOT AccountLocked

This means all the good conditions must be true, and the bad condition must be false.

A reliable method for any GCSE truth table

When you see a truth table question, slow down and use columns.

1. Write the input columns.
2. List all possible input combinations.
3. Add intermediate columns for smaller parts of the expression.
4. Fill each intermediate column using one operator at a time.
5. Fill the final output column.