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Before we learn the implementation of full adder using half adder first we learn two important Combinational Logic Circuits known as the Half Adder Circuit and the Full Adder Circuit.

Half adder is a combinational logic circuit perform addition of two single bit number. It is a digital circuit has two input X, Y and two output sum (S), carry (C). Here sum is the least significant bit (LSB) and carry is the most significant bit (MSB). Half adder circuit is used in computer ALU (Arithmetic and Logic Unit ) to perform arithmetic operation.

Now, we find the Boolean logical expression from truth table for outputs Sum (S) and carry (C) and draw circuit diagram.

Sum = X’Y+XY’ = X xor Y
Carry = XY

Full adder is a combinational logic circuit perform addition of three single bit number. It is a digital circuit has three inputs A, B and Cin , where Cin is the previous carry and two output sum (S), carry (Cout).

Now, we find the Boolean logical expression from truth table for outputs Sum (S) and carry (C) and draw circuit diagram.

Sum =A xor B xor Cin

Co = AB + ACin +BCin or Cin  (A ⊕ B) +AB

A full adder can be implemented by logically connecting two half adders and OR gate.

From earlier calculations, we get the equations for Sum (S) and Carry (Cout ) of a Full Adder :

S = A’ B’ Cin + A’ B C’in  + A B’ C’in  + A B Cin

Cout = A B + A Cin  + B Cin

Now, we can rewrite the equation for Sum Output as under:

S = A’ B’ Cin + A’ B C’in + A B’ C’in + A B Cin

= Cin (A’ B’ + A B) + C’in (A’ B + A B’)

=Cin (A Ex-NOR B) + C’in (A Ex-OR B)

= Cin (A ⊕ B)’ + C’in (A ⊕ B)

Therefore, S = Cin ⊕ (A ⊕ B) =  A ⊕ B ⊕ Cin

Now, we write the expression for carry output Cout :

CO = A B + A Cin + B Cin

= A B + A Cin+ B Cin (A + A’)

= A B + A Cin + A B Cin + A’ B Cin

= A B (1 + Cin)+ A Cin + A’ B Cin

= A B + A Cin + A’ B Cin

= A B + A Cin (B + B’) + A’ B Cin

= A B + A B Cin + A B’ Cin + A’ B Cin

= A B (1 + Cin )+Cin (A B’ + A’ B)

= A B + Cin (A B’ + A’ B)

= A B + Cin (A ⊕ B)

Therefore, CO = A B + Cin (A ⊕ B)

The above expression of Sum and Carry output is same that for a full adder. Therefore, we have showed that circuit shown in fig 1.3, really act like a Full Adder.

Now, for more detail watch my video

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