The Two Boolean States

Digital systems work with signals that are treated as being in one of two states. These states are represented by the binary digits 0 and 1. In Boolean logic, 0 normally means false, off, inactive, not detected or not satisfied. The value 1 normally means true, on, active, detected or satisfied.

The meaning of 0 and 1 depends on the definition in the question. A value of 1 does not always mean that a device is physically switched on. For example, an input called WaterLow may be defined as 1 when the water level is below a safe limit. The important step is to read the given definition rather than assume a meaning.

A logic circuit receives one or more binary inputs and applies gate rules to produce a binary output. The output can then control a device such as a warning lamp, buzzer, motor or lock. In examination questions the physical device is often described, but the circuit itself is analysed using 0 and 1.

Representing Real Conditions As Inputs

A problem statement normally identifies conditions that can be tested. Each condition is assigned a variable, often a single capital letter. For example, D may represent whether a door is open, A may represent whether an alarm is armed, and X may represent whether the siren sounds.

The question should define when each input equals 1. A clear definition might be: D = 1 when the door is open; A = 1 when the alarm is armed. If the door is closed, D = 0. If the alarm is not armed, A = 0.

Do not confuse the variable name with its value. D is the name of the input. At one moment D may equal 0, and at another moment it may equal 1.

Inputs, Intermediate Results And Outputs

A simple circuit may connect inputs directly to one gate. A more complex circuit may produce intermediate results. For example, inputs A and B may enter an AND gate and produce an intermediate value P. Then P and input C may enter an OR gate to produce output X.

Intermediate variables are useful when tracing a circuit or constructing a truth table. They allow each gate to be evaluated separately. The final output must still be one binary value because the syllabus limits logic circuits to one output.

The maximum circuit size required by the syllabus is three inputs and one output. A circuit can contain several gates, but each gate except NOT is limited to two inputs.

Reading A Problem Statement

The words used in a statement often indicate a gate operation. The phrase both A and B indicates AND. The phrase A or B, including the possibility that both are true, indicates OR. The phrase either A or B but not both indicates XOR. The phrase not A indicates NOT.

NAND and NOR are usually identified as the opposite result of AND or OR. For example, an output that is 0 only when both inputs are 1 follows NAND. An output that is 1 only when both inputs are 0 follows NOR.

When the statement contains brackets or separate clauses, determine the smaller conditions first. The order of operations must be shown by the circuit and the expression.

A Basic Input And Output Table

Boolean Logic Does Not Describe Voltage Detail

At this level, the two binary states are used as logical values. Candidates do not need to calculate electronic voltages or study transistor construction. The focus is the relationship between input combinations and the output produced by the gates.

A circuit diagram should therefore be interpreted as a logical process. Follow the values through the gates in the direction of the signal arrows or wires, from the inputs on the left towards the output on the right.

From Condition To Decision

A practical system can be described in five stages: sensors or switches detect conditions; the conditions are represented by input values; gates process those values; a binary output is produced; and the output controls a device.

For example, a security circuit can use D = 1 when a door is open and A = 1 when the alarm is armed. An AND rule can make X = 1 only when both conditions are true. The siren is then activated by X = 1.

Worked Examples

Examination Guidance

• Write the meaning of each variable before analysing the circuit.
• Use the definitions given in the question; do not invent the opposite meaning for 1.
• Separate physical descriptions from their binary representations.
• Remember that all required circuits have no more than three inputs and one output.
• Show intermediate values when a circuit contains more than one gate.

Knowledge Check

1. What two values are used in Boolean logic?

2. What does 1 usually represent?

3. Can NOT have two inputs?

4. What is the syllabus limit for a circuit?

5. Why are intermediate variables useful?