← IGCSE Computer Science Paper 2

Checking that a circuit, expression and truth table agree

A circuit, expression and truth table are equivalent when each gives the same output X for the same inputs. Check by working out X row by row in each one and comparing the results.

Part of: IGCSE Computer Science 0478 — 10 Boolean Logic — Module 39

Understanding Checking that a circuit, expression and truth table agree in IGCSE Computer Science Paper 2

What it does: The circuit inverts A before the AND gate but the expression leaves out the NOT. For A=1, B=1 the circuit gives 0 but the expression gives 1. Following the signal through each gate in turn produces the circuit's X for that row, which is then checked against the truth table row. Equivalent means the same output X for the same inputs, checked row by row. A AND B = 0, then 0 OR C (1) gives 1. 1 AND 1 = 1, so the expression and the table disagree. The table is not equivalent to the expression. One row that differs proves they are not equivalent, but you need every row to match to prove they agree.

Common mistakes: With A=1 and B=0, A OR B is 1 because OR outputs 1 when at least one input is 1. The line should be A OR B = 1. AND outputs 1 only when both inputs are 1. With A=1 and B=1, A AND B = 1, so the line should be A AND B = 1.

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This module's key terms are also in a printable crossword (Modules 562–580) and word search (Modules 539–580).

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Learning
Duo
A circuit, an expression and a truth table are equivalent. What must be true?
They give the same X for every input rowThey use the same number of gates
Matches
Expression: X = NOT (A AND B). Circuit: NOT A, then AND B. The student says they agree.
↓
The brackets are in the wrong place. NOT (A AND B) inverts the result of the AND, but the circuit inverts only A before the AND, so they give different outputs.
Applying
Meanings
Term
equivalent
Meaning
A circuit, expression and truth table are equivalent when each gives the same output X for the same inputs. Check by working out X row by row in each one and comparing the results.
Practice
X = (A AND B) OR C
Testing
Gaps
X =
Check Answer
Spots
2Row: A=1 B=0 C=1
3A OR B = 0

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Quick Questions

1. A circuit, an expression and a truth table are equivalent. What must be true?
A. They give the same X for every input row ✓
B. They use the same number of gates
Equivalent means the same output X for the same inputs, checked row by row.
2. X = (A AND B) OR C with A=1, B=0, C=1. What is X?
A. X = 1 ✓
B. X = 0
A AND B = 0, then 0 OR C (1) gives 1.
3. A truth table shows X=0 for A=1, B=1. The expression is X = A AND B. Do they agree on this row?
A. Yes, both give X=0
B. No, the expression gives X=1 for this row ✓
1 AND 1 = 1, so the expression and the table disagree. The table is not equivalent to the expression.
4. One matching row is enough to prove a circuit and a truth table agree.
A. False: every row must match ✓
B. True: one matching row proves it
One row that differs proves they are not equivalent, but you need every row to match to prove they agree.
5. A circuit has a NOT gate on A feeding an AND gate with B. Its expression is written X = A AND B. Do they match?
A. Yes, a NOT gate does not change the output
B. No, the NOT on A is missing from the expression ✓
The expression should be X = (NOT A) AND B. A missing NOT changes the output on some rows.

Match the Pairs

X = (A AND B) OR C → Evaluate this by working out A AND B first, then OR the result with C. Do this for every row of the truth table.
A=1, B=0, C=1 → X=? → A test row: put these input values into the circuit and the expression, then compare the two outputs X.
A missing NOT in an expression makes it differ from the circuit on some rows, so the two are not equivalent → Circuit: NOT A, then AND B | Expression: X = A AND B
Tracing a circuit row by row gives an output X that you can compare with the matching row of the truth table → A=1, B=1: A AND B = 1, NOT 1 = 0, so X = 0
Expression: X = NOT (A AND B). Circuit: NOT A, then AND B. The student says they agree. → The brackets are in the wrong place. NOT (A AND B) inverts the result of the AND, but the circuit inverts only A before the AND, so they give different outputs.
The student checks only the row A=0, B=0 and says the circuit and the truth table agree. → One row is not enough. Two different circuits can match on one row, so you must test every row, or at least rows where the outputs could differ.

Key Term

equivalent
A circuit, expression and truth table are equivalent when each gives the same output X for the same inputs. Check by working out X row by row in each one and comparing the results.

Fill in the Blanks

1. A = 1, B = 0, C = 1 X = (A AND B) OR C X = 1
Hint: Work out the output for this row: do the bracket first, then the OR. Type the value of X.
2. Circuit: B OR C first, then AND with A X = A AND (B OR C)
Hint: The OR must be worked out first, so type the symbol that opens the bracket round it.
3. Table: A=1,B=0 gives X=1; A=1,B=1 gives X=0 X = A AND NOT B
Hint: The expression has a missing gate name. Test both table rows to find which gate is needed in front of B.
4. Inputs A, B and C: test all combinations Number of rows in the truth table: 8
Hint: Type how many rows are needed to test every combination of three inputs.
5. Circuit, expression, table: same X per row They are equivalent
Hint: Type the word for logic that always gives the same output for the same inputs.

Spot the Error

Question 1
A X = (A OR B) AND C
B Row: A=1 B=0 C=1
C A OR B = 0
D X = 1 AND 1 = 1
Line C — With A=1 and B=0, A OR B is 1 because OR outputs 1 when at least one input is 1. The line should be A OR B = 1.
Question 2
A X = NOT (A AND B)
B Row: A=1 B=1
C A AND B = 0
D X = NOT 1 = 0
Line C — AND outputs 1 only when both inputs are 1. With A=1 and B=1, A AND B = 1, so the line should be A AND B = 1.
Question 3
A Circuit: NOT A, then OR B
B Expression: X = (NOT A) AND B
C Row A=0 B=1: NOT 0 = 1
D 1 AND 1 = 1, so X = 1
Line A — The circuit must match the expression (NOT A) AND B, so the second gate is an AND gate, not an OR gate. The line should be 'Circuit: NOT A, then AND B'.
Question 4
A X = A OR (NOT B)
B Row: A=0 B=1
C NOT B = 0
D X = 0 OR 0 = 1
Line D — OR outputs 0 when both inputs are 0. So 0 OR 0 = 0, not 1. The line should be X = 0 OR 0 = 0.
Question 5
A Table row: A=1 B=0 gives X=0
B Expression: X = A AND B
C 1 AND 0 = 1
D Both give 0, so they agree
Line C — AND needs both inputs to be 1. With 1 AND 0 the output is 0, which is what the table row shows. The line should be 1 AND 0 = 0.

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