Boolean Algebra with example

These are a few examples of how we can use Boolean Algebra to simplify larger digital logic circuits.

Example 1

Construct a Truth Table for the logical functions at points C, D and Q in the following circuit and rewrite the single logic gate that can be used to replace the whole circuit.

 

The circuit consists of a 2-input NAND gate, a 2-input EX-OR gate and finally a 2-input EX-NOR gate at the output(Q). As there are only 2 inputs to the circuit labelled A and B, there can only be 4 possible combinations of the input ( output will be combination of those inputs =22 ). Plotting the logical functions from each gate in tabular form will give us the following truth table for the whole of the logic circuit below.

Inputs		Output at
A	B	C	D	Q
0	0	1	0	0
0	1	1	1	1
1	0	1	1	1
1	1	0	0	1

In this truth table, column C represents the output function generated by the NAND gate, while column D represents the output function from the Ex-OR gate. Both of these two output expressions then become the input  for the Ex-NOR gate.

By looking at the final output at Q we can say, the whole of the above circuit can be replaced by just one single 2-input OR Gate.

Example 2

Find out the Boolean algebra expression for the following circuit.

 

The system consists of an AND Gate(A.B), a NOR Gate(A+B) and finally an OR Gate(A+B). Both these expressions are also separate inputs to the OR gate. Thus the final output expression is given as:

 

The output of the circuit is given as Q = (A.B) + (A+B), but the notation A+B is the same as the De Morgan´s notation A.B, Then substituting A.B into the output expression gives us a final output notation of Q = (A.B)+(A.B), which is the Boolean notation for an Exclusive-NOR Gate as seen in the previous section.

Inputs		Intermediates		Output
B	A	A.B	A + B	Q
0	0	0	1	1
0	1	0	0	0
1	0	0	0	0
1	1	1	0	1

Then, the whole circuit above can be replaced by just one single Exclusive-NOR Gate and indeed an Exclusive-NOR Gate is made up of these individual gate functions.