## Adder:

Adder is the main part of computer’s ALU (Arithmetic and Logic Unit). With the help of which we perform all types of Arithmetical and Logical operations. But ALU never works with decimal numbers, that’s why two or more binary numbers can be added with this type of logic circuit. Adder is binary summator. So you need to know the general rules of binary addition. For example 0+0=00, 0+1=01, 1+0=01, 1+1=11. In this case the sum is shown in two bits where the left bit is carry and the right bit is sum. **Adder is of two types**. Namely –

- Half Adder
- Full Adder

## Half Adder:

Half Adder is the simplest combinational logic circuit, with the help of which we can add two binary bits (x and y). Sum and carry are calculated by adding two bits together with the help of half adder. Half Adder truth table is shown –

In this case the Boolean equations of sum and carry are –

Sum = x⊕y = xy + xy; Carry= x.y Therefore

Sum is x XOR y; Carry is x AND y

So the logic circuit for the logic gate of Half Adder is the combination of XOR gate for Sum (S) and AND gate for Carry (C).

### From the truth table we get:

S (Sum) = 1 then,

When x = 0 and y = 0 from the truth table

∴ S=x.y [Remember: If x or y has value 0 (zero), bar ( ‾‾ ) sits on its head]

Again S (Sum) = 1,

When x = 1 and y = 0 from the truth table

∴ S = x.y

So, S = x.y + x.y [The two are OR (+).]

S(Sum) = x⊕y

Again, looking at carry or C, we find that C=1 if x=1 and y=1.

∴ C = x.y

## Full Adder:

Full Adder is a combinational logic circuit that adds three or more bits. Like half adder, full adder and three bits are added together to calculate the sum and carry. Full Adder will have a carry bit (C_{i} called carry in) in addition to two bits x and y as input and S (Sum) and C_{0} carry out as output. Full Adder truth table is shown-

### From the truth table we get:

S (Sum )= 1, when

x=0 y=0 C_{i}=1 [i.e xyC_{i}]

x=0 y=1 C_{i}=0 [i.e xyC_{i}]

x=1 y=0 C_{i}=0 [i.e xyC_{i}]

x=1 y=1 C_{i}=1 [i.e xyC_{i}]

i.e S= x yC_{i}
+
xyC_{i}
+
x yC_{i}
+
xyC_{i}

i.e C_{0} (Carry Out) = 1 when will,

x=0 y=1 C_{i}=1 [i.e xyC_{i}]

x=1 y=0 C_{i}=1 [i.e xyC_{i}]

x=1 y=1 C_{i}=0 [i.e xyC_{i}]

x=1 y=1 C_{i}=1 [i.e xyC_{i}]

∴ C_{0}(Carry out) = xyC_{i} + xyC_{i} + xyC_{i} + xyC_{i}

From the truth table, S (Sum) and C_{0} (Carry out) are Boolean equations –

S= x yC_{i}
+
xyC_{i}
+
x yC_{i}
+
xyC_{i}

= x(yC_{i} + yC_{i}) + x(y C_{i} + yC_{i})

= x(y⊕C_{i}) + x(y⊕C_{i})

=x⊕y⊕C_{i} [Sum = x XOR y XOR C_{i} ]

Again, C_{0}= xyC_{i} + xyC_{i} +xyC_{i} + xyC_{i}

C_{0}= x(yC_{i} + yC_{i}) + x(yC_{i} + yC_{i})

= C_{i}(xy + xy) + xy (C_{i} + C_{i}) [we know A + A = 1]

=C_{i} (x⊕y) + xy

Then we can make the logic circuit of Full Adder from Boolean equation is-

In fact, two half adders are connected consecutively and form a full adder with the help of an OR gate. The logic diagram of the adder is shown-

## Frequently Asked Question (FAQs):

- What is Adder?
- Half Adder.
- Full Adder.