Minimize circuit functions

Question

$\begin{array}{rrrr | rr } 0& 0 & 0 & 0 & 1 &1 &1 &1 &1 & 1&0 \\ 0& 0 & 0 & 1 & 0 &1 &1 &0 &0 & 0&0 \\ 0& 0 & 1 & 0 & 1 &1 &0 &1 &1 & 0&1 \\ 0& 0 & 1 & 1 & 1 & 1&1 &1 &0 & 0&1 \\ 0& 1 & 0 & 0 & 0 &1 & 1&0 &0 & 1&1 \\ 0& 1 & 0 & 1 & 1 &0 &1 &1 &0 & 1&1 \\ 0& 1 & 1 & 0 & 1 &0 &1 &1 &1 & 1&1 \\ 0& 1 & 1 & 1 & 1 &1 &1 &0 &0 & 0&0 \\ 1& 0 & 0 & 0 & 1 &1 &1 &1 &1 & 1&1 \\ 1& 0 & 0 & 1 & 1 &1 &1 &1 &0 & 1&1 \\ \end{array}$

Can someone explain me how to minimize this function?

right side is my output I know how to build the functions but not how to minimize functions with this type of scale

Answer 1

First you have to build the formula whose behavior is mirrored in the truth table, for large circuits it is impossible to simplify simply by looking at the truth table. Once you have the formula, you can proceed to apply the axioms and rules of Boolean algebra to shrink the formula.

Examples of Rules:

1. $\lnot(\lnot A \lor B) \lor A = A$

2. $\lnot(\lnot A \lor B) \lor \lnot(\lnot A \lor \lnot B)=A$

3. $A+AB=A$ (Algebraic manipulation)

Answer 2

What is exactly ur problem?

I mean do u need help understanding Karnaugh maps, or expressions reductions???

Karnaugh Maps for O/P one:

u draw a table for the i/p values say $AB$ for the rows and $CD$ for the columns, but remember that $11$ precedes $10$ to simplify the reduction. $d$ means $Don't$ $Care$ to use the non existing values to simplify the reduction.

C'D'     C'D      CD   CD'

A'B' 1 0 1 1

A'B 0 1 1 1

AB d d d d

AB' 1 1 d d

To simplify the F1 we take as large squares as we can (circulation is allowed)

-We take the 2 last columns together making $D$

+The 2 last rows making $A$

Then we have 2 remaining 1s, for the 1st of them we take the middle square:

• $BC$

We have the 1st corner remaining 1, trying to largen any square or rectangle, we use the $circular$ reduction and take the 4corners. i.e.,

• $B'D'$

Thus, F1=A+D+BC+B'D'

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Now, example of the expression reduction method on the last O/P... say F7

We write down the expression for the each 1 in F7, then add them together (add here represents OR) and try to simplify normally. F7 has 1s at: 2,3,4,5,6,8,9

F7= A'B'CD'+ A'B'CD + A'BC'D' + A'BC'D+ A'BCD' + AB'C'D' + AB'C'D

Now simplify...

F7=(A'B'CD'+ A'B'CD) + (A'BC'D' + A'BC'D) + A'BCD' + (AB'C'D' + AB'C'D)

F7= A'B'C(D'+D) + A'BC'(D'+D) + A'BCD' + AB'C'(D' +D)

= A'B'C+ (A'BC'+ A'BCD') + AB'C'
= A'B'C+ A'B(C'+CD') + AB'C'

and so on,...

(u see, the karnaugh map is usually easier)