MHB How to use K-map to prove a boolean equation?

[FallArk]

This is actually a computer engineering problem:
How to use k-map to show that B + AB’C’D + AB’CD = B + AD?
Isn't K-map a graph that shows combinations of input A, B, C and D?
What is B then?

 

[I like Serena]

This is actually a computer engineering problem:
How to use k-map to show that B + AB’C’D + AB’CD = B + AD?
Isn't K-map a graph that shows combinations of input A, B, C and D?
What is B then?

Hey FallArk! ;)

A k-map or Karnaugh map is not a graph - it's a table that looks like:

It's a method to simplify boolean expressions of a number of boolean variables (4 variables in the example).

B is just one of the 4 boolean variables.
The AB at the top represents all possible combinations of the 2 boolean variables A and B.
And the CD at the left represents all possible combinations of C and D.
If we make such tables for both the left hand side and the right hand side, we should find that the tables are identical.

 

[FallArk]

Hey FallArk! ;)

A k-map or Karnaugh map is not a graph - it's a table that looks like:It's a method to simplify boolean expressions of a number of boolean variables (4 variables in the example).

B is just one of the 4 boolean variables.
The AB at the top represents all possible combinations of the 2 boolean variables A and B.
And the CD at the left represents all possible combinations of C and D.
If we make such tables for both the left hand side and the right hand side, we should find that the tables are identical.

Sorry for not wording it correctly, what I was trying to say is that how can I tell where to fill in the 1s and 0s in the map according to what is given to me?

 

[I like Serena]

Sorry for not wording it correctly, what I was trying to say is that how can I tell where to fill in the 1s and 0s in the map according to what is given to me?

The rightmost 2 columns correspond to $A=1$, or just $A$.
The center columns correspond to $B$.

We have an expression that begins with $B + ...$.
That means that it's true if $B$ is.
So we can fill in all fields corresponding to $B=1$.

\begin{array}{c|c|c|c|}
{}_{\small{CD}}\backslash{}^{\small{AB}}&\tiny{00}&\tiny{01}&\tiny{11}&\tiny{10} \\
\hline
\tiny{00}&\ & 1&1&\ \\
\hline
\tiny{01}&\ & 1&1&\ \\
\hline
\tiny{11}&\ & 1&1&\ \\
\hline
\tiny{10}&\ & 1&1&\ \\
\hline
\end{array}

Now we need to add 1's for the other parts of the expression.

For instance $AD$ is represented by:
\begin{array}{c|c|c|c|}
{}_{\small{CD}}\backslash{}^{\small{AB}}&\tiny{00}&\tiny{01}&\tiny{11}&\tiny{10} \\
\hline
\tiny{00}& & & & \\
\hline
\tiny{01}& & & 1& 1\\
\hline
\tiny{11}& & & 1&1 \\
\hline
\tiny{10}& & & & \\
\hline
\end{array}

So $B+AD$ is:
\begin{array}{c|c|c|c|}
{}_{\small{CD}}\backslash{}^{\small{AB}}&\tiny{00}&\tiny{01}&\tiny{11}&\tiny{10} \\
\hline
\tiny{00}&0 &1 &1 &0 \\
\hline
\tiny{01}&0 &1 &1 &1\\
\hline
\tiny{11}&0 &1 &1 &1 \\
\hline
\tiny{10}&0 &1 &1 &0 \\
\hline
\end{array}

 

[FallArk]

The rightmost 2 columns correspond to $A=1$, or just $A$.
The center columns correspond to $B$.

We have an expression that begins with $B + ...$.
That means that it's true if $B$ is.
So we can fill in all fields corresponding to $B=1$.

\begin{array}{c|c|c|c|}
{}_{\small{CD}}\backslash{}^{\small{AB}}&\tiny{00}&\tiny{01}&\tiny{11}&\tiny{10} \\
\hline
\tiny{00}&\ & 1&1&\ \\
\hline
\tiny{01}&\ & 1&1&\ \\
\hline
\tiny{11}&\ & 1&1&\ \\
\hline
\tiny{10}&\ & 1&1&\ \\
\hline
\end{array}

Now we need to add 1's for the other parts of the expression.

For instance $AD$ is represented by:
\begin{array}{c|c|c|c|}
{}_{\small{CD}}\backslash{}^{\small{AB}}&\tiny{00}&\tiny{01}&\tiny{11}&\tiny{10} \\
\hline
\tiny{00}& & & & \\
\hline
\tiny{01}& & & 1& 1\\
\hline
\tiny{11}& & & 1&1 \\
\hline
\tiny{10}& & & & \\
\hline
\end{array}

So $B+AD$ is:
\begin{array}{c|c|c|c|}
{}_{\small{CD}}\backslash{}^{\small{AB}}&\tiny{00}&\tiny{01}&\tiny{11}&\tiny{10} \\
\hline
\tiny{00}&0 &1 &1 &0 \\
\hline
\tiny{01}&0 &1 &1 &1\\
\hline
\tiny{11}&0 &1 &1 &1 \\
\hline
\tiny{10}&0 &1 &1 &0 \\
\hline
\end{array}

I get it now! So I just have to make sure all the 1's I fill in corresponds to what it given. Thanks!