A1.2.4 Construct and analyse truth tables.

The big idea

A truth table is a complete, explicit map from every possible combination of Boolean input values (0 / 1 or F / T) to the corresponding output of a logical condition, expression or circuit. Because this map is exhaustive, it lets us predict, verify and simplify digital logic with mathematical certainty. Truth tables therefore sit at the centre of computer architecture, programming language semantics and formal reasoning.

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1 Constructing & analysing truth tables

Typical workflow	What you are doing	Why it matters
List all input combinations	For n inputs write the 2ⁿ binary rows in Gray or binary order.	Guarantees completeness.
Compute the output(s)	Apply gate rules or Boolean algebra row-by-row.	Turns a diagram or expression into concrete data.
Interpret / analyse	Look for functional patterns (e.g., rows where output = 1).	Exposes redundancy, hazards, or useful canonical forms.

1.1 From simple gates to a table

For the circuit Y = (A ⋅ ¬B) + (¬A ⋅ B) (an XOR built from AND, OR, NOT):

A	B	¬A	¬B	A⋅¬B	¬A⋅B	Y
0	0	1	1	0	0	0
0	1	1	0	0	1	1
1	0	0	1	1	0	1
1	1	0	0	0	0	0

The table predicts that the output is high exactly when the inputs differ—confirming XOR behaviour.

1.2 From problem description to table

“A warning LED must light only when the door is open or the lock sensor is faulty, but not when both happen.”

Translate to Boolean:
LED = Door_open ⊕ Lock_fault ⇒ same XOR table above, but using domain-specific input names.

1.3 Relating tables ↔ Boolean expressions

• Construct (command term): derive the table row-by-row from the expression.
• Analyse (command term): read the last column and re-express it.
  Good answer: “Rows 2 and 3 are the only 1s, so the minimal sum-of-products is A̅B + AB̅, i.e., XOR.”
  Less-good answer: “I see two 1s so it’s some kind of OR thing.”

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2 Deriving tables from logic diagrams

1. Label every internal wire.
2. Write Boolean sub-expressions gate-by-gate.
3. Populate the truth table using those sub-expressions.
4. Isolate the final output column for later simplification.

This systematic decomposition is vital when a diagram contains fan-out, shared subcircuits or mixed-level logic.

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3 Using truth tables to simplify logic

3.1 Algebraic simplification

• Identify common factors row-wise or use identities (e.g., consensus theorem) directly on the derived Boolean expression.

3.2 Karnaugh maps (K-maps)

1. Copy the 1-rows of the truth table into a 2-D Gray-coded grid.
2. Group adjacent 1-cells in powers of two (1, 2, 4, 8…).
3. Write a product term for each group (variables that stay constant).
4. Sum the terms to get the minimal sum-of-products.

For the XOR example, the K-map forms two separate groups containing single 1-cells, yielding the same minimal expression A̅B + AB̅.

Why K-maps matter: They reveal opportunities that are hard to spot algebraically, especially with four to six inputs, and they lead directly to fewer gates—critical for chip area, power and delay.

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4 Command-term spotlight

Command term	Examiner expects	Good student response	Weak response
Construct	Produce the full table with correct headings, all 2ⁿ rows, and output values.	“Table lists 16 rows for 4 inputs; each output value computed with De Morgan where needed.”	“I sketched some rows and guessed the rest.”
Analyse	Extract meaning, e.g., minimal expression, detection of equivalence or hazard.	“Analysis shows outputs true when inputs have odd parity; circuit implements XOR.”	“Looks random; maybe it’s OR?”

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Key take-aways

• A truth table is the canonical behavioural description of any combinational logic block.
• Constructing one demands precision; analysing one demands insight.
• Boolean algebra and Karnaugh mapping convert the raw table into an implementable, economical circuit.
  Master these steps and you have the toolkit to move seamlessly between problem statements, equations, diagrams and hardware implementations.