Boolean Logic

Logic gates

NOT

The NOT logic gate is the only logic gate with one input; the rest have two.

The truth table, which is used to trace the output of logic gates, shows that with input A, it gives output X.

AND

OR

NAND (NOT AND)

NOR (NOT OR)

XOR

Logic expressions, logic circuits, problem statements and truth tables

Writing a logic expression for a logic circuit

• Logic gate 1: NOT A
• Logic gate 2: NOT C

To write down the expressions for the other gates, we will have to combine the expressions of the previous gates.

• Logic gate 3 (logic gate 1 and B): NOT A AND B
• Logic gate 4 (logic gates 3 and 2): (NOT A AND B) OR NOT C

X = (NOT A AND B) OR NOT C

Completing a truth table from a logic circuit

This is how you solve a truth table:

1. Count the number of gates and make the required number of columns in the working space (columns 1, 2 and 3).
2. Write the output of each gate one by one. This is where your knowledge of the output of different gates will be useful.
3. The final output will be written in the column labelled X.

Drawing a logic circuit from a logic expression

This is how you make a logic circuit from a logic expression:

1. Solve the brackets first.
2. Remember that the NOT gate only has one input and the others have two.
3. Be careful about the positions of the brackets in the expression.
4. The dots represent joints. They mean that the same value is to be used somewhere else (here B goes to both the NAND and the XOR).

Writing a logic expression from a truth table

X = (NOT A AND NOT B AND NOT C) OR (A AND NOT B AND C) OR (A AND B AND C)

Tips to solve this type of question:

• a. The final expression will only consist of the parts where X (the final output) is 1.
• b. Every time X is 1, A, B and C will be connected using AND.
• c. Each output where X is 1 will be connected using OR.
• d. When an input is 1, it will be written as A. When it is 0, it will be written as NOT A.

Building a circuit from a problem statement

Create a logic circuit that will produce an output (X) of 1 when:

• the switch S1 is on, and
• either P1 is less than 3 or P2 is less than 3, but not both.

All logic gates used must have a maximum of two inputs.

How to solve:

• a. The word in the middle always represents the last logic gate. In this case, "and" represents the AND gate.
• b. Secondly, the question states that we need to have an output of 1. Since we already know that the AND gate only gives an output of 1 when both the inputs are 1, we need to make sure both the inputs are 1.
• c. When the switch S1 is ON, it gives an output of 1, hence we connect it directly to the AND gate.
• d. The question states that either P1 or P2 needs to be less than 3 but not both. By recalling what we have learnt before, we recognise that this is the output achieved by the XOR gate.