[10.3a] Logic gates & logic circuits

Logic gates & logic circuits

From statement, expression, or truth table - building circuits without simplification

This page shows how to translate three common forms of Boolean logic descriptions into a corresponding logic circuit: a written problem statement, a symbolic logic expression, and a truth table. You must draw circuits exactly as specified, without simplification. All examples are limited to at most three inputs and a single output, matching the assessment constraints.

In IGCSE Computer Science you should be comfortable with the core logic gates: NOT (inverter), AND, OR, plus the derived gates NAND, NOR, and XOR where needed. Each gate takes binary signals (0 for false/low, 1 for true/high) and produces a binary output. When converting between formats, be precise about the order of operations, any negations, and the exact wiring that matches the description.

Starter: symbols and reading order

• Inputs are usually labelled A, B, C. The single output is labelled Q or X.
• NOT is shown as a small triangle with a bubble. In expressions use an overbar or apostrophe (e.g. A' or ¬A).
• AND and OR combine signals. Use brackets in expressions to remove ambiguity, e.g. (A AND B) OR C. Without brackets, evaluation order must be explicitly given in the question.
• Truth tables list every input combination as 0/1 rows and the corresponding output for each row.

Three pathways to the same destination

Below are three worked examples of the same general skill: creating a logic circuit from different starting points. Use the tabs to switch between them and notice the common patterns in identifying inputs, operators, and any required inversions.

• Problem Statement
• Logic Expression
• Truth Table

Context: A sprinkler system turns on (F) only if:

• The soil is dry (A = 1),
• It is not raining (B = 0), and
• It is daytime (C = 1)

Boolean Expression: F = A AND (NOT B) AND C

Description: All three conditions must be met: dry soil, no rain, and daytime for the sprinkler to activate.

The Boolean expression F = (A AND B) OR NOT C can be represented as:

F represents the Output. Any letter is OK, eg O.
Often, expressions are written simply as: (A AND B) OR NOT C. You decide on the letter to represent the Output.

Given truth table: produce a circuit with one output

Suppose the required output Q for inputs A, B is defined as follows:

A	B	Q
0	0	0
0	1	1
1	0	1
1	1	0

This pattern is XOR: output 1 when inputs are different. To build it directly from the table without simplifying, you can use a sum-of-products approach:

1. Identify the rows where Q=1: rows 01 and 10.
2. For each such row, create an AND term that matches the row exactly:
   • Row A=0, B=1 → (¬A AND B)
   • Row A=1, B=0 → (A AND ¬B)
3. OR the two terms to get Q: Q = (¬A AND B) OR (A AND ¬B).

Wire it literally: two NOT gates for A and B as required, two AND gates, then one OR gate to combine them.

Method: step-by-step translation checklist

1. Name inputs clearly (e.g. sensors, switches). Keep to at most three inputs.
2. Spot negations like "not", "unless", "except" which imply a NOT gate.
3. Group conditions with brackets mentally, matching the grammar of the sentence or the given expression.
4. Choose gate order to honour the grouping. The wiring diagram is the physical version of the expression.
5. Verify by checking a few input combinations against your statement/expression/table.

Common gate behaviours (for recall)

Deep Dive: Sum-of-Products and Product-of-Sums (without simplifying)

When you are given a truth table, a reliable way to build a circuit is to form a sum-of-products (SOP). For every row where Q=1, create an AND term that matches that row exactly, using NOT where an input is 0. Then OR all these terms together. The dual form, product-of-sums (POS), creates an OR term for each row where Q=0 and ANDs the terms. Although more advanced courses might simplify these expressions, in this benchmark you should keep the direct form to mirror the given table precisely.

Edge cases you might encounter

• Constant output: If the table shows Q=0 for all rows, the circuit is simply a fixed low output. If Q=1 for all rows, it is a fixed high output. Draw what is requested.
• Ignored input: A statement may define an output that does not depend on one of the inputs. Do not add unnecessary gates - leave that input disconnected from the logic path.
• Three-input gates: If a term requires three inputs, you may use a 3-input gate symbol if allowed, or build it from 2-input gates by cascading (e.g. AND A with B, then AND the result with C).