Construct truth tables for simple logic circuits using combinations of…

The idea

To construct a truth table for a simple circuit, first identify every input and list all input combinations in binary counting order: two inputs give 4 rows (00, 01, 10, 11), three inputs give 8 rows. Add a working column for each gate output, working from the inputs towards the final output. Apply each gate's rule: NOT inverts its single input; AND gives 1 only when all its inputs are 1; OR gives 1 when at least one input is 1; XOR gives 1 only when its inputs differ. For example, for A AND (NOT B), compute NOT B first, then AND it with A. Finally read off the output column. Check each row against the gate rules rather than guessing, and confirm the number of rows equals 2ⁿ for n inputs.

What earns the marks

• List all input combinations in binary order, with 2ⁿ rows for n inputs.
• Include a separate column for each intermediate gate output as well as the final output.
• Apply NOT by inverting the single input value.
• Apply AND as 1 only when all inputs are 1.
• Apply OR as 1 when at least one input is 1.
• Apply XOR as 1 only when the inputs differ.
• Evaluate gates in the correct order from inputs to final output.

Common mistakes and how to put them right

• Omitting input combinations: the learner lists only some rows.

  Put it right: For n inputs there must be 2ⁿ rows, e.g. 4 rows for two inputs.

• Skipping intermediate columns: the learner jumps straight to the final output and makes errors.

  Put it right: Add a working column for each gate output.

• Applying AND as if it were OR: the learner gives 1 when only one input is 1. Correction: AND requires all inputs to be 1.

• Evaluating gates in the wrong order.

  Put it right: Work from the inputs through each gate towards the final output.

Exam tips

• Count the inputs first and write 2ⁿ rows before filling any values.
• Add labelled working columns for each gate to reduce mistakes.
• Recheck the final column row by row against each gate rule.