Logic Construction (Cambridge (CIE) A Level Computer Science) : Revision Note
From problem statements
A problem statement is typically a written interpretation of a scenario that requires a specific logical outcome
Logic circuits can be constructed using from problem statements
Worked Example
A server room has an automatic cooling fan (F)
The fan turns on based on the following input parameters:
Parameter | Description | Binary Value | Condition |
|---|---|---|---|
T | Temperature | 1 = Too high | 0 = Acceptable |
H | Humidity | 1 = Too high | 0 = Acceptable |
M | Maintenance mode | 1 = Active | 0 = Inactive |
D | Door sensor | 1 = Door open | 0 = Door closed |
The fan (F = 1) turns on if:
The temperature is too high
AND humidity is too high
AND the door is closed
UNLESS maintenance mode is active (if maintenance is active, the fan must stay off)
Draw a logic circuit to represent how the fan (F) operates based on the input conditions. [3]
Answer
NOT gates on D and M inputs [1 mark]
AND gate combines T AND H or NOT D AND NOT M [1 mark]
Combines two previous AND outputs (e.g. (T AND H) AND (NOT D AND NOT M)) [1 mark]
From logic expressions
A logic expression is a way of showing how a logic circuit works using symbols and Boolean logic (AND, OR, NOT)
It describes the conditions that must be met for the output to be true (1) or false (0)
From a logic expression, a logic circuit and/or truth table can be constructed
Worked Example
A logic expression is given:
S = (A AND B AND C) OR (B XOR C)
(a) Draw the logic circuit for the given expression [4]
(b) Complete the truth table for the logic expression: [2]
S = (A AND B AND C) OR (B XOR C)
Answer
(a)
Each correct gate [1 mark]
(b)
A | B | C | S |
|---|---|---|---|
0 | 0 | 0 | 0 |
0 | 0 | 1 | 1 |
0 | 1 | 0 | 1 |
0 | 1 | 1 | 0 |
1 | 0 | 0 | 0 |
1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 |
Rows 1 to 4 correct [1 mark]
Rows 5 to 8 correct [1 mark]
From truth tables
To create a truth table for the expression P = (A AND B) AND NOT C
Calculate the numbers of rows needed (2number of inputs)
In this example there are 3 inputs (A, B, C) so a total of 8 rows are needed (23)
To not miss any combination of inputs, start with 000 and count up in 3-bit binary (0-7)
A | B | C |
|---|---|---|
0 | 0 | 0 |
0 | 0 | 1 |
0 | 1 | 0 |
0 | 1 | 1 |
1 | 0 | 0 |
1 | 0 | 1 |
1 | 1 | 0 |
1 | 1 | 1 |
Add a new column to show the results of the brackets first (A AND B)
A | B | C | A AND B |
|---|---|---|---|
0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 1 | 0 | 0 |
0 | 1 | 1 | 0 |
1 | 0 | 0 | 0 |
1 | 0 | 1 | 0 |
1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 |
Add a new column to show the results of NOT C
A | B | C | A AND B | NOT C |
|---|---|---|---|---|
0 | 0 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 0 | 1 |
0 | 1 | 1 | 0 | 0 |
1 | 0 | 0 | 0 | 1 |
1 | 0 | 1 | 0 | 0 |
1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 1 | 0 |
The last column shows the result of the Boolean expression (P) by comparing (A AND B) AND NOT C //
A | B | C | A AND B | NOT C | P |
|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 0 | 0 |
1 | 1 | 0 | 1 | 1 | 1 |
1 | 1 | 1 | 1 | 0 | 0 |
Examiner Tips and Tricks
It is possible to create a truth table when combining expressions that show only the inputs and the final outputs.
The inclusion of the extra columns supports the process but can be skipped if you feel able to do those in your head as you go.
From logic circuits
From a logic circuit you can create the logic expression and/or truth table
Worked Example
Write the logic expression for the given logic circuit. [3]
Answer
(A NAND B) OR (B XOR C)
A NAND B [1 mark]
B XOR C [1 mark]
OR [1 mark]
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