Boolean logic

What you'll learn

• What Boolean values are and how they are written as 1 and 0.
• How to recognise and use the AND, OR and NOT logic gate symbols.
• How to complete truth tables for single gates and combined logic diagrams.
• How to turn simple real-world conditions into Boolean expressions and logic diagrams.

Boolean values: true or false

Computer systems often make decisions using conditions such as:

• “Is the password correct?”
• “Is the file available?”
• “Has the button been pressed?”

Each condition has only two possible states: yes or no, true or false, 1 or 0.

You may also see alternatives such as T/F for true/false, or symbols such as ∨ for OR. OCR accepts other valid notation, but in these notes we will use 1, 0, AND, OR and NOT.

Logic gates

A logic gate is a diagram symbol that takes one or more Boolean inputs and produces one Boolean output.

For OCR J277, you need to know three gates:

• AND, also called conjunction
• OR, also called disjunction
• NOT, also called negation

The symbols and truth tables below are worth learning carefully.

The AND gate

An AND gate outputs 1 only if all its inputs are 1.

For example, a door might unlock only if:

• A = the card is valid
• B = the PIN is correct

The output is 1 only when both conditions are true.

The OR gate

An OR gate outputs 1 if at least one of its inputs is 1.

For example, a warning light might turn on if:

• A = the temperature is too high
• B = the pressure is too high

Only one of these problems needs to be true for the warning light to turn on.

The NOT gate

A NOT gate has one input and reverses it.

So if A is 1, NOT A is 0. If A is 0, NOT A is 1.

The NOT symbol is a triangle with a small circle at the output end. The small circle shows that the signal is being inverted.

Truth tables

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

For two inputs, use four rows:

For three inputs, use eight rows. In general, for $n$ inputs, the number of rows is $2^n$.

When a logic diagram has more than one gate, it is usually easiest to add intermediate columns. These are extra columns for the outputs of gates before the final output.

Combining Boolean operators

You can combine gates to make more complex decisions. A combined logic diagram is read by following the wires from the inputs to the final output.

Here is a diagram for the expression Q = (A AND NOT B) OR C.

The expression has three stages:

1. B is passed through a NOT gate, giving NOT B.
2. A and NOT B go into an AND gate, giving A AND NOT B.
3. That result is ORed with C, giving the final output Q.

Creating logic diagrams from scenarios

In exam questions, you may be given a scenario in words and asked to create, complete or edit a logic diagram or truth table.

A good method is:

1. Identify each condition that can be true or false.
2. Give each condition a clear input letter, such as A, B or C.
3. Choose the correct gate:
   • Use AND when all conditions must be true.
   • Use OR when at least one condition must be true.
   • Use NOT when a condition must be false.
4. Combine gates step by step.

Editing incomplete truth tables or diagrams

Sometimes a question gives you a partly completed diagram or truth table. You may need to fill in missing gates, wires or outputs.

Useful patterns:

• If the output is 1 only when both inputs are 1, the gate is AND.
• If the output is 1 when either input is 1, the gate is OR.
• If the output is always the opposite of the input, the gate is NOT.

If there are brackets in a Boolean expression, use them to decide which gates are connected first. For example, (A OR B) AND C is different from A OR (B AND C).