Checking that a circuit, expression and truth table agree

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Checking that a circuit, expression and truth table agree

Name:
Date:
Class:
A

Quick Questions

Shade in the A or B circle for each question.

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
2. X = (A AND B) OR C with A=1, B=0, C=1. What is X?
A X = 1
B X = 0
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
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
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
B

Match the Pairs

Draw a line connecting each item on the left to its match on the right.

Left columnRight column
1X = (A AND B) OR C AOne 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.
2A=1, B=0, C=1 → X=? BCircuit: NOT A, then AND B | Expression: X = A AND B
3A missing NOT in an expression makes it differ from the circuit on some rows, so the two are not equivalent CEvaluate this by working out A AND B first, then OR the result with C. Do this for every row of the truth table.
4Tracing a circuit row by row gives an output X that you can compare with the matching row of the truth table DA=1, B=1: A AND B = 1, NOT 1 = 0, so X = 0
5Expression: X = NOT (A AND B). Circuit: NOT A, then AND B. The student says they agree. EA test row: put these input values into the circuit and the expression, then compare the two outputs X.
6The student checks only the row A=0, B=0 and says the circuit and the truth table agree. FThe 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.
C

Definitions

equivalent
D

Fill in the Blanks

Complete each line of code by filling in the missing parts.

1.
A = 1, B = 0, C = 1 X = (A AND B) OR C X = _
2.
Circuit: B OR C first, then AND with A X = A AND _B OR C)
3.
Table: A=1,B=0 gives X=1; A=1,B=1 gives X=0 X = A AND ___ B
4.
Inputs A, B and C: test all combinations Number of rows in the truth table: _
5.
Circuit, expression, table: same X per row They are __________
E

Spot the Error

Shade in the letter of the line that contains an 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
Question 2
A X = NOT (A AND B)
B Row: A=1 B=1
C A AND B = 0
D X = NOT 1 = 0
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
Question 4
A X = A OR (NOT B)
B Row: A=0 B=1
C NOT B = 0
D X = 0 OR 0 = 1
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

Answer Key

Checking that a circuit, expression and truth table agree
Section A: Quick Questions
Q1
A — They give the same X for every input row
Q2
A — X = 1
Q3
B — No, the expression gives X=1 for this row
Q4
A — False: every row must match
Q5
B — No, the NOT on A is missing from the expression
Section B: Match the Pairs
Q1
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.
Q2
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.
Q3
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
Q4
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
Q5
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.
Q6
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.
Section C: Definitions
Q1
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.
Section D: Fill in the Blanks
Q1
A = 1, B = 0, C = 1 X = (A AND B) OR C X = 1
Q2
Circuit: B OR C first, then AND with A X = A AND (B OR C)
Q3
Table: A=1,B=0 gives X=1; A=1,B=1 gives X=0 X = A AND NOT B
Q4
Inputs A, B and C: test all combinations Number of rows in the truth table: 8
Q5
Circuit, expression, table: same X per row They are equivalent
Section E: Spot the Error
Q1
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.
Q2
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.
Q3
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'.
Q4
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.
Q5
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.