The output Y of a 2-bit comparator is logic 1 whenever the 2-bit input A is greater than the 2-bit input B. The number of combinations for which the output is logic 1, is
6
Comparator Function Explained
A 2-bit comparator is a digital circuit that compares two 2-bit binary numbers, say input A and input B, and determines their relationship (i.e., whether A > B, A < B, or A = B). In this specific problem, we are interested in finding the number of input combinations for which the output Y is logic 1, which occurs when input A is strictly greater than input B (A > B).
Input Combinations for 2-bit Numbers
Since both input A and input B are 2-bit numbers, each can represent values from 0 to 3 in decimal. Let's denote the 2-bit input A as \(A_1A_0\) and input B as \(B_1B_0\). The possible decimal values for A and B are:
- 00 (0 in decimal)
- 01 (1 in decimal)
- 10 (2 in decimal)
- 11 (3 in decimal)
The total number of possible input combinations for (A, B) is \(4 \times 4 = 16\).
Determining Logic 1 Output (A > B)
We need to list all possible combinations of A and B and identify the cases where A is greater than B. For these cases, the output Y will be logic 1. Let's examine each possible value of A and see which values of B make A greater than B.
| Input A (Decimal) | Input A (Binary) | Input B (Decimal) | Input B (Binary) | Condition (A > B) | Output Y |
|---|---|---|---|---|---|
| 0 | 00 | 0 | 00 | False | 0 |
| 0 | 00 | 1 | 01 | False | 0 |
| 0 | 00 | 2 | 10 | False | 0 |
| 0 | 00 | 3 | 11 | False | 0 |
| 1 | 01 | 0 | 00 | True | 1 |
| 1 | 01 | 1 | 01 | False | 0 |
| 1 | 01 | 2 | 10 | False | 0 |
| 1 | 01 | 3 | 11 | False | 0 |
| 2 | 10 | 0 | 00 | True | 1 |
| 2 | 10 | 1 | 01 | True | 1 |
| 2 | 10 | 2 | 10 | False | 0 |
| 2 | 10 | 3 | 11 | False | 0 |
| 3 | 11 | 0 | 00 | True | 1 |
| 3 | 11 | 1 | 01 | True | 1 |
| 3 | 11 | 2 | 10 | True | 1 |
| 3 | 11 | 3 | 11 | False | 0 |
Counting Combinations for Logic 1 Output
From the table above, we can count the number of instances where the condition A > B is true, and consequently, the output Y is logic 1:
- When A = 1 (decimal), B must be 0 (decimal). (1 combination: (A=01, B=00))
- When A = 2 (decimal), B can be 0 or 1 (decimal). (2 combinations: (A=10, B=00), (A=10, B=01))
- When A = 3 (decimal), B can be 0, 1, or 2 (decimal). (3 combinations: (A=11, B=00), (A=11, B=01), (A=11, B=10))
Summing these up:
Total combinations for Y = 1 = 1 + 2 + 3 = 6.
Therefore, there are 6 combinations for which the output of the 2-bit comparator is logic 1 (A > B).
Important Questions from Comparators
X = X1X0 and Y = Y1Y0 are 2-bit binary numbers. The Boolean function S that satisfies the condition “If X > Y, then S = 1”, in its minimized form, is
