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Section 5.6 Exercises

1.

Design a function that will detect the even 4-bit integers.

Solution

Let a 4-bit integer be \(wxyz\) where each literal represents one bit. The even 4-bit integers are given by the function:

\begin{alignat*}{1} F(w,x,y,z) \amp {}={} w' \cdot x' \cdot y' \cdot z' + w' \cdot x' \cdot y \cdot z' + w' \cdot x \cdot y' \cdot z' + w' \cdot x \cdot y \cdot z' \\ \amp \quad + w \cdot x' \cdot y' \cdot z' + w \cdot x' \cdot y \cdot z' + w \cdot x \cdot y' \cdot z' + w \cdot x \cdot y \cdot z' \end{alignat*}

Using the distributive property repeatedly we get:

\begin{alignat*}{1} F(w,x,y,z) \amp {}={} z' \cdot (w' \cdot x' \cdot y' + w' \cdot x' \cdot y + w' \cdot x \cdot y' + w' \cdot x \cdot y \\ \amp \quad + w \cdot x' \cdot y' + w \cdot x' \cdot y + w \cdot x \cdot y' + w \cdot x \cdot y) \\ \amp {}={} z' \cdot (w' \cdot (x' \cdot y' + x' \cdot y + x \cdot y' + x \cdot y) \\ \amp \quad + w \cdot (x' \cdot y' + x' \cdot y + x \cdot y' + x \cdot y)) \\ \amp {}={} z' \cdot (w' + w) \cdot (x' \cdot y' + x' \cdot y + x \cdot y' + x \cdot y) \\ \amp {}={} z' \cdot (w'+ w) \cdot (x' \cdot (y' + y) + x \cdot (y' + y)) \\ \amp {}={} z' \cdot (w' + w) \cdot (x' + x) \cdot (y' + y) \end{alignat*}

And from the complement property we arrive at a minimal sum of products:

\begin{gather*} F(w,x,y,z) = z' \end{gather*}

which you recognize as Figure 5.1.3.

2.

Find a minimal sum of products (mSoP) expression for the function

\begin{align*} F(x,y,z) &= x' \cdot y' \cdot z' + x' \cdot y' \cdot z + x' \cdot y \cdot z'\\ &\quad+\ x \cdot y' \cdot z' + x \cdot y \cdot z' + x \cdot y \cdot z \end{align*}
Solution

First we draw the Karnaugh map:

Several groupings are possible. Keep in mind that groupings can wrap around. We will work with:

which yields a minimal sum of products:

\begin{equation*} F(x,y,z) = z' + x' \cdot y' + x \cdot y \end{equation*}
3.

Find a minimal product of sums (mPoS) expression for the function

\begin{align*} F(x,y,z) &= (x + y + z) \cdot (x + y + z') \cdot (x + y' + z')\\ &\quad \cdot \ (x' + y + z) \cdot (x' + y' + z') \end{align*}
Solution

This expression includes maxterms 0, 1, 3, 4, and 7. These appear in a Karnaugh map:

Next we encircle the largest adjacent blocks, where the number of cells in each block is a power of two. Notice that maxterm \(M_{0}\) appears in two groups.

From this Karnaugh map it is very easy to write the function as a minimal product of sums:

\begin{equation*} F(x,y,z) = (x + y) \cdot (y + z) \cdot (y' + z') \end{equation*}
4.

Find a minimal product of sums (mPoS) expression for the function

\begin{align*} F(x,y,z) &= x' \cdot y' \cdot z' + x' \cdot y' \cdot z + x' \cdot y \cdot z'\\ &\quad+\ x \cdot y' \cdot z' + x \cdot y \cdot z' + x \cdot y \cdot z \end{align*}
Solution

Using the Karnaugh map zeros:

we obtain the complement of our desired function:

\begin{equation*} F'(x,y,z) = x' \cdot y \cdot z + x \cdot y' \cdot z \end{equation*}

and from DeMorgan's Law:

\begin{equation*} F(x,y,z) = (x + y' + z') \cdot (x' + y + z') \end{equation*}
5.

Show where each minterm is located with this Karnaugh map axis labeling using the notation of Figure 5.5.7.

6.

Show where each minterm is located with this Karnaugh map axis labeling using the notation of Figure 5.5.7.

7.

Design a logic function that detects the prime single-digit numbers. Assume that the numbers are coded in 4-bit BCD (see Section 4.6.1). The function is \(1\) for each prime number.

Solution

The prime numbers correspond to the minterms \(m_{2}\text{,}\) \(m_{3}\text{,}\) \(m_{5}\text{,}\) and \(m_{7}\text{.}\) The minterms \(m_{10}\text{,}\) \(m_{11}\text{,}\) \(m_{12}\text{,}\) \(m_{13}\text{,}\) \(m_{14}\text{,}\) \(m_{15}\) cannot occur so are marked “don't care” on the Karnaugh map.

which gives:

\begin{equation*} F(w,x,y,z) = x \cdot z + x' \cdot y \end{equation*}