Combining Boolean Operators (OCR GCSE Computer Science)

• Boolean operators can be combined to produce more complex expressions
• The combination of two or more Boolean operators forms a logic expression
• Brackets are used to clarify the order of operations
• A logic diagram is a visual representation of combinations of Boolean operators within a logic circuit
• An example would be Q = NOT(A OR B)

• In the diagram above, the inputs are represented by A and B
• P is the output of the OR gate on the left and becomes the input of the NOT gate
• Q is the final output of the logic circuit
• This is a logic diagram for the Boolean expression Q = NOT(A OR B)

Example of combining Boolean operators

• P = (A OR B) AND NOT C

What is a truth table?

• A truth table is a tool used in logic and computer science to visualise the results of Boolean expressions
• They represent all possible inputs and the associated outputs for a given Boolean expression
• To create a truth table for the expression P = (A AND B) AND NOT C
  • Calculate the numbers of rows needed (2^{number 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 by comparing (A AND B) AND NOT C

A	B	C	A AND B	NOT C	(A AND B) AND NOT C
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