How do we write logic as Boolean expressions, simplify them, and build any gate from NAND gates?
Boolean expressions and the basic Boolean identities, simplifying combinational logic, building any logic function from NAND gates (NAND universality), and designing a logic system to meet a requirement using a data sheet to select ICs.
A focused answer to WJEC Eduqas GCSE Electronics on Boolean algebra and NAND logic, covering Boolean expressions and identities, simplifying logic, building any gate from NAND gates, and designing a logic system using a data sheet.
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What this topic is asking
WJEC Eduqas wants you to write combinational logic as Boolean expressions, use the basic Boolean identities to simplify logic, understand that NAND gates can build any logic function (so a circuit can be reduced to NAND gates), and design a logic system to meet a stated requirement, using a data sheet to choose a suitable logic IC.
Boolean expressions
Boolean notation is a compact way to write what a logic circuit does without drawing it. You can read an expression straight off a circuit by following the gates: an AND gate makes a product, an OR gate a sum, a NOT gate adds a bar. Each Boolean expression has a matching truth table, and you can move freely between the expression, the truth table and the gate diagram.
Boolean identities and simplification
Simplifying matters because every gate costs money, space and power. For example, is just , so any gate implementing it is unnecessary; is always 1, so that part of a circuit can be replaced by a fixed logic 1. Reducing the number of gates is a real design benefit, and questions reward spotting that an expression can be simplified.
NAND universality (redundancy)
NAND universality is a key practical idea. A NAND with its inputs tied together inverts (acts as NOT); a NAND followed by a NAND-as-inverter makes an AND; and combinations of NANDs make OR. Because only NAND gates are needed, a designer can build a circuit from one chip type, which simplifies stock, layout and testing. (NOR gates are universal too, but NAND is the usual example.) This is sometimes called NAND redundancy because the other gate types become unnecessary.
Designing a logic system
Design works from the specification to the circuit. State exactly when the output must be 1, capture that as a truth table or Boolean expression, simplify to cut the gate count, and pick real logic ICs. The data sheet gives the chip's supply voltage range, how many gates it contains, the current its outputs can supply, and the input and output logic levels. Choosing parts within these limits, and adding a transistor or relay to drive any high-current output, completes the design.
Try this
Q1. Simplify the Boolean expression . [1 mark]
- Cue. Anything AND 0 is 0, so .
Q2. State how a NAND gate can be made to act as a NOT gate. [1 mark]
- Cue. Connect both inputs of the NAND together (to the single signal), and it inverts.
Exam-style practice questions
Practice questions written in the style of WJEC exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
Eduqas style3 marksWrite the Boolean expression for the output of a two-input AND gate followed by a NOT gate, and name the single gate that gives the same function.Show worked answer →
A Component 1 Boolean question. An AND gate of inputs A and B gives ; passing this through a NOT gate inverts it to (2 marks for the AND term and the overall inversion bar). The single gate that gives is a NAND gate (1 mark). Markers reward the AND product, the inversion of the whole expression, and naming the NAND gate. A common error is to invert each input separately (), which is a different function.
Eduqas style4 marksA burglar alarm output must sound (logic 1) only when the system is armed AND a sensor is triggered. Design the logic, give the Boolean expression, and explain how you would choose the logic IC.Show worked answer →
A Component 1 design question. The output sounds only when both conditions are true, which is an AND function of the two inputs, armed (A) and triggered (T): (2 marks for identifying AND and the Boolean expression). The circuit is a single two-input AND gate driving the alarm through a transistor (1 mark). To choose the IC, read a data sheet to find an AND-gate chip with the right supply voltage and output current, and check it can drive the next stage (1 mark). Markers reward the AND function, the Boolean expression, and using a data sheet to pick a suitable IC. A common error is to use OR, which would sound when only one condition is met.
Related dot points
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Sources & how we know this
- WJEC Eduqas GCSE Electronics specification (from 2017) — WJEC Eduqas (2017)