Given function f(C, B, A) = CA + BA, expand it to represent it in its canonical SOP form.

Question

Accepted Answer

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 [latex](C \bar{C})[/latex], since [latex](C \bar{C})[/latex] = 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 [latex](B+ \bar{B})[/latex] to the original function, which will not alter the initial logic of the function because [latex](B+\bar{B})[/latex] = 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 [latex](B\bar{B})[/latex] 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 [latex](C \bar{C}[/latex]) with the term BA. Hence:

[latex]f(C, B, A)=C A(B \bar{B})+(C \bar{C}) B A .[/latex] (7.9)

Applying logic product distribution and making sure that variables are consistently organized from C down to A (e.g., CBA)

[latex]f(C, B, A)=C B A+C \bar{B} A+C B A+\bar{C} B A[/latex] (7.10)

Eliminating only the second instance of the term CBA because it is redundant yields:

[latex]f(C, B, A)=C \bar{B} A+C B A+\bar{C} B A[/latex] (7.11)

Rewriting Equation (7.11) in SOP form and rearranging terms:

[latex]f(C, B, A)= m _5+ m _7+ m _3=\sum(3,5,7) .[/latex] (7.12)

Question 7.8: Given function f(C, B, A) = CA + BA, expand it to represent ...

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