Numerical Representation
Product of Sums Representation
Numerical Representation
| Decimal Value | A | B | C | f |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 |
| 2 | 0 | 1 | 0 | 1 |
| 3 | 0 | 1 | 1 | 1 |
| 4 | 1 | 0 | 0 | 0 |
| 5 | 1 | 0 | 1 | 0 |
| 6 | 1 | 1 | 0 | 0 |
| 7 | 1 | 1 | 1 | 1 |
The function has a value 1 for the combinations shown, therefore:
......(1)Note that the summation sign indicates that the terms
are "OR'ed" together. The function can be further reduced to the form:
(000, 010, 011, 111)It is self-evident that the binary form of a function can be written directly from the truth table.
Note:
It follows from the last expression that the binary form can
be replaced by the equivalent decimal form, namely:
(0,2,3,7)......(2)
Product of Sums Representation
From the truth table given above the function has the value 0 for
the combinations shown, therefore
......(3)
Writing the inverse of this function:
Applying De Morgan's Theorem we obtain:
Applying the second De Morgan's Theorem we obtain:
......(4)The function is expressed in standard product of sums form.
Thus there are two forms of a function, one is a sum of products form (either standard or normal) as given by expression (1), the other a product of sums form (either standard or normal) as given by expression (4). The gate implementation of the two forms is not the same!
Examples
In binary form: f(A, B, C, D) = (0101, 1011, 1100, 0000, 1010, 0111)
In decimal form: f(A, B, C, D) = (5, 11, 12, 0, 10, 7)
Problem
