Question

g A four-input digital circuit is used to detect whether the resulting product of two two-bits numbers is greater or equal than 3. Consider that the two-bit numbers are given as the inputs BA and DC. So, your design will implement a function X (D, C, B, A) that is "0" if the product of BA X DC is less than 3 and "1" if it is greater or equal than 3. For example if your inputs are (D, C, B, A) = 1010, then the result value is "1" (X (D, C, B, A) =1), since BA =10 (210 ) and DC =10 (210)and the product will be 4, which is greater or equal than 3. On the other hand, if for example, (D, C, B, A) = 1001 then (X (D, C, B, A)=0), since BA =01 (110) and DC =10 (210) with a product equals to 2. The minterm expression for the function X (D, C, B, A) is:

Answer 1

Tthe minterm expression for the function X is:

X = B'ADC + BA'DC' + BA'DC + BAD'C + BADC' + BADC

(The character ' refers to the complement of the variable)

Explanation:

It is important to know that a minterm refers to a product of all variables that have a result of 1 in the truth table. The variables can be with or without complement. For example, consider the following truth table:

A, B, F

0, 0, 0

0, 1, 0

1, 0, 0

1, 1, 1

According to the previous truth table, the minterm would the row that F equals 1. In this case, when A and B are 1, F is 1. So, the variables are used directly (without complement). The minterm is: F = AB

Now, according to the problem, the truth table is presented below. The idea is to complete the X column which is '0' if the product BA x DC is less than 3 and is '1' if the product BA x DC is greater or equal than 3.

• B, A, D, C, X

1. 0, 0, 0, 0, 0 (0x0=0)
2. 0, 0, 0, 1, 0 (0x1=0)
3. 0, 0, 1, 0, 0 (0x2=0)
4. 0, 0, 1, 1, 0 (0x3=0)
5. 0, 1, 0, 0, 0 (1x0=0)
6. 0, 1, 0, 1, 0 (1x1=1)
7. 0, 1, 1, 0, 0 (1x2=2)
8. 0, 1, 1, 1, 1 (1x3=3)
9. 1, 0, 0, 0, 0 (2x0=0)
10. 1, 0, 0, 1, 0 (2x1=2)
11. 1, 0, 1, 0, 1 (2x2=4)
12. 1, 0, 1, 1, 1 (2x3=6)
13. 1, 1, 0, 0, 0 (3x0=0)
14. 1, 1, 0, 1, 1 (3x1=3)
15. 1, 1, 1, 0, 1 (3x2=6)
16. 1, 1, 1, 1, 1 (3x3=9)

The idea is to get the rows that have '1' in the X column. The minterms are are: 8, 11, 12, 14, 15 and 16. The minterm is composed by the four variables. To indicate a '1' just put the letter. To indicate a '0' put a letter with this character: '. For example, the row 8 would be B'ADC, because B is '0' and the others are '1'. The total minter expression is:

X = 'minterm 8' + 'minterm 11' + 'minterm 12' + 'minterm 14' + 'minterm 15' + 'minterm 16'

X = B'ADC + BA'DC' + BA'DC + BAD'C + BADC' + BADC

Thus, the minterm expression for the function X is:

X = B'ADC + BA'DC' + BA'DC + BAD'C + BADC' + BADC