The operation of a single bit digital comparator can be expressed as a truth table:

InputsOutputs

A	B	A > B	A = B	A < B
0	0	0	1	0
0	1	0	0	1
1	0	1	0	0
1	1	0	1	0

The operation of a two bit digital comparator cannot be expressed as a truth table:

Inputs Outputs

A1	A0	B1	B0	A < B	A = B	A > B
0	0	0	0	0	1	0
0	0	0	1	1	0	0
0	0	1	0	1	0	0
0	0	1	1	1	0	0
0	1	0	0	0	0	1
0	1	0	1	0	1	0
0	1	1	0	1	0	0
0	1	1	1	1	0	0
1	0	0	0	0	0	1
1	0	0	1	0	0	1
1	0	1	0	0	1	0
1	0	1	1	1	0	0
1	1	0	0	0	0	1
1	1	0	1	0	0	1
1	1	1	0	0	0	1
1	1	1	1	0	1	0

Following is the circuit digram for the two bit comparator :

Implementation

A = A₃A₂A₁A₀

B = B₃B₂B₁B₀

Here each subscript represents one of the digits in the numbers.

Equality

The binary numbers A and B will be equal if all the pairs of significant digits of both numbers are equal, i.e.,

A₃ = B₃, A₂ = B₂, A₁ = B₁ and A₀ = B₀

Since the numbers are binary, the digits are either 0 or 1 and the boolean function for equality of any two digits Ai and Bi can be expressed as

x_i is 1 only if A_i and B_i are equal.

For the equality of A and B, all x_i variables (for i=0,1,2,3) must be 1.

So the quality condition of A and B can be implemented using the AND operation as

(A = B) = x₃x₂x₁x₀

The binary variable (A=B) is 1 only if all pairs of digits of the two numbers are equal.

Inequality

In order to manually determine the greater of two binary numbers, we inspect the relative magnitudes of pairs of significant digits, starting from the most significant bit, gradually proceeding towards lower significant bits until an inequality is found. When an inequality is found, if the corresponding bit of A is 1 and that of B is 0 then we conclude that A>B.

This sequential comparison can be expressed logically as:

(A>B) and (A < B) are output binary variables, which are equal to 1 when A>B or A<B respectively.