Cambridge O Level Computer Science · Syllabus 2210 · Boolean Logic
Truth Table
What is Truth Table?
A table that lists every possible combination of the inputs to a logic gate or logic circuit, one combination per row in binary counting order, together with the binary output produced for each combination; a table with n inputs has 2 to the power n rows and may carry extra columns for intermediate gate outputs.
This definition is part of the Boolean Logic chapter in Cambridge O Level Computer Science.
Truth Table in context
One logic problem can be written in four ways: as a problem statement in words, as a logic expression in letters and word operators, as a logic circuit of gate symbols and wires, and as a truth table listing the output for every possible input combination. Topic 10 is, almost entirely, the skill of moving between them. The syllabus lists nine such moves, and each one is a question that can be asked.
A truth table shows the output of a gate or a circuit for every possible combination of its inputs. Not the likely combinations, not the sensible ones — every one. That completeness is the whole point: once a truth table is finished, the behaviour of the circuit is fully described and there is nothing left to find out.
Completing a truth table from a circuit is the most reliably examinable skill in Topic 10, and it is also the one where a good method is worth the most marks. The method is: give every gate output a column, fill the columns in dependency order, and never write a value in a later column before the columns it depends on are filled.
Completing a truth table from an expression is the same job as completing one from a circuit, and for a good reason: they are the same thing written two ways. Each operator is a column; the brackets tell you the order the columns go in.
Going straight from a problem statement to a truth table skips the expression, and that makes one step more important than any other: write down what 1 means for every input before you fill in a single cell. Almost every error in this conversion is an input whose meaning quietly flipped half way down the table.
Given a truth table, you can always write an expression that reproduces it, using a routine that needs no algebra at all. Look only at the rows where the output is 1. Each of those rows becomes one AND term, and the terms are joined with OR. Nothing is simplified, and nothing more advanced than the six gates is used.
Once you can get from a truth table to an expression, getting to a circuit is just section 10.3 C again. The sequence is fixed, and the last step — verifying against the table you started from — is the one that catches everything.
Everything in Topic 10 that looks like cleverness is a trap. If a question hands you a structure — as a statement, as an expression or as a truth table — your job is to reproduce that structure. An answer that produces identical outputs from a different arrangement of gates has answered a question that was not asked.
Common mistakes with Truth Table
- “OR means exactly one of the inputs is 1.”ActuallyThe OR gate is inclusive: it outputs 1 when one input is 1, and also when both are. 1 OR 1 = 1.The real gate“Exactly one” is XOR, and it is a different symbol with a different truth table.Why it happensEveryday English uses “or” exclusively — “tea or coffee” rarely means both.
- “XOR and EOR are two different gates.”ActuallyThey are two names for one gate. The syllabus writes it as XOR (EOR). Same symbol, same truth table, same everything.In the examUse whichever name the question uses; neither is worth more.
- “A truth table can leave out combinations that could not happen in real life.”ActuallyA truth table lists every possible combination of the inputs, whether or not the physical system could produce it.WhyThe table describes the logic, not the plumbing. A tank sensor pair that could never read 0 and 1 together still gets that row.
- “A three-input truth table has six rows.”ActuallyIt has eight. Rows are 2 to the power n: two inputs give four rows, three inputs give eight.Fast checkIf the row count is not 2, 4 or 8, something is missing.
- “The brackets in an expression are optional.”ActuallyThe brackets are the grouping. (T OR H) AND (NOT R) and T OR (H AND (NOT R)) are different circuits with different truth tables.Habit to buildBracket every part you write, even when you think it is obvious.
- “You need Boolean algebra laws and De Morgan’s laws for Topic 10.”ActuallyNone of them appears anywhere in this syllabus. Neither do half adders, flip-flops, sequential logic, timing diagrams or transistor circuits.What Topic 10 does needSix symbols, six truth tables, and the nine conversions. That is the whole of it.
Examiner tips on Truth Table
- “Neither” and “not both” have two acceptable shapes. Which one you should write depends on the question. If it simply asks for an expression from a statement, either form is correct as long as the truth table matches. If the question has already given you a structure, or a later part asks you to draw the circuit for the expression you wrote, keep the two forms straight: A NOR B is one gate, NOT (A OR B) is two, and the drawing must match whichever you wrote.
Questions students ask about Truth Table
How many rows does a truth table need?
The number of rows is 2 raised to the power of the number of inputs: one input gives 2 rows, two inputs give 4 rows, and three inputs give 8 rows, listed in binary counting order. If a row count is not 2, 4 or 8, a combination has been missed.
Are Boolean algebra laws or Karnaugh maps needed for this topic?
No. Neither Boolean algebra laws, De Morgan's laws nor Karnaugh maps appear anywhere on this syllabus, because they exist to simplify circuits, and this topic explicitly forbids simplification. What is needed is the six gate symbols, their truth tables, and the conversions between statement, expression, circuit and truth table.