Combination Circuit

In Digital logic we have two kinds of circuits -Combinational Logic Circuit and Sequential logic circuit.

Combinational logic circuit are those where output is determined by the circuit itself i.e only by its inputs.In combinational circuit output depends upon only the pure function of input. the part of an arithmetic logic unit, or ALU, that does mathematical calculations is constructed using combinational logic. Other circuits used in computers, such ashalf adders, full adders, half subtractors, full subtractors, multiplexers, demultiplexers, encoders and decoders are also made by using combinational logic.

Sequential Circuit are those which has a memory element and output depends not only upon the inputs but also the state or content of memory element.Basically the memory stores the value of previous output of the circuit.

Mostly used Combinational Circuits-

Half-Adder:
To design a half adder we have to think what the adder will do.Let's see it through a truth table-
       

A	B	S	C
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

                           
Here the inputs are A and B.Their sum is S and as S can hold only one bit so carry C.How from truth table we can get the formula for S and C is discussed before.
so S=AB'+A'B and C=AB
For the Half-Adder circuit click here  
NAND gate implementation of half-adder-

Half-Adder with NAND gate

Full Adder:

A half-adder has only two inputs and  there is no  provision to add carry from lower order bits when multibit addition is performed.So another input is included which is the carry from previous add operation.So with the carry the truth table will look like this-

Inputs			Outputs
An	Bn	C(n-1)	Sn	Cn
0	0	0	0	0
0	0	1	1	0
0	1	0	1	0
0	1	1	0	1
1	0	0	1	0
1	0	1	0	1
1	1	0	0	1
1	1	1	1	1

From the truth table the function for Sn=An XOR Bn XOR C(n-1)

                                                       Cn=AnBn+(An XOR Bn)Cn

The Sn and Cn function is obtained from Karnaugh Map.Apply to the table above, you'll get the functions.

So The corresponding full adder circuit will look like this-

Full Adder Circuit

Half and Full Subtractor:

Half Subtractor truth table given below to understand the