Question 7.8: Given function f(C, B, A) = CA + BA, expand it to represent ...
Given function f(C, B, A) = CA + BA, expand it to represent it in its canonical SOP form.
The given function clearly is not already given as a sum of its minterms.
What we need to do is to create “logic redundancies” that do not affect the original logic of the function. For example, ANDing terms like (C \bar{C}), since (C \bar{C}) = 1 and ANDing 1 to any logical expression does not alter its original logic, is a way of creating such redundancy. Another type of possible redundancy is ANDing terms like (B+ \bar{B}) to the original function, which will not alter the initial logic of the function because (B+\bar{B}) = 1. Proceeding with our function f:
f (C, B, A) = CA+ BA. (7.8)
Since (Eq. 7.8) term CA is missing the literal B we AND the term (B\bar{B}) with the term CA without changing the original logic of function f. At the same time we create a redundancy to the term BA by ANDing the term (C \bar{C}) with the term BA. Hence:
f(C, B, A)=C A(B \bar{B})+(C \bar{C}) B A . (7.9)
Applying logic product distribution and making sure that variables are consistently organized from C down to A (e.g., CBA)
f(C, B, A)=C B A+C \bar{B} A+C B A+\bar{C} B A (7.10)
Eliminating only the second instance of the term CBA because it is redundant yields:
f(C, B, A)=C \bar{B} A+C B A+\bar{C} B A (7.11)
Rewriting Equation (7.11) in SOP form and rearranging terms:
f(C, B, A)= m _5+ m _7+ m _3=\sum(3,5,7) . (7.12)