Problem 4 (10 points) Simplify the following Boolean expression, ($\overline{A}B$)(AB) + AB, then implement the simplified result using NAND gates. (Hint: Boolean identities in Lecture 12.)
Added by Rebecca H.
Close
Step 1
$F = \overline{A}AB + AB$ $F = A\overline{A}B + AB$ Using the Boolean identity $X \cdot \overline{X} = 0$, we have $A\overline{A} = 0$. $F = 0 \cdot B + AB$ $F = 0 + AB$ $F = AB$ Show more…
Show all steps
Your feedback will help us improve your experience
Kanak Aggarwal and 71 other AP CS educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Simplify the Boolean expression $(\overline{A \cdot \bar{B}+C}) \cdot(\bar{A}+\overline{B \cdot \bar{C}})$ by using de Morgan's laws and the rules of Boolean algebra. pplying de Morgan's laws to the first term gives: $$ \begin{aligned} \overline{A \cdot \bar{B}+C} &=\overline{A \cdot \bar{B}} \cdot \bar{C}=(\bar{A}+\overline{\bar{B}}) \cdot \bar{C} \\ &=(\bar{A}+B) \cdot \bar{C}=\bar{A} \cdot \bar{C}+B \cdot \bar{C} \end{aligned} $$ pplying de Morgan's law to the second term gives: $$ \bar{A}+\overline{B \cdot \bar{C}}=\bar{A}+(\bar{B}+\overline{\bar{C}})=\bar{A}+(\bar{B}+C) $$ hus $(\overline{A \cdot \bar{B}+C}) \cdot(\bar{A}+\overline{B \cdot \bar{C}})$ $$ \begin{aligned} &=(\bar{A} \cdot \bar{C}+B \cdot \bar{C}) \cdot(\bar{A}+\bar{B}+C) \\ &=\bar{A} \cdot \bar{A} \cdot \bar{C}+\bar{A} \cdot \bar{B} \cdot \bar{C}+\bar{A} \cdot \bar{C} \cdot C \\ &\quad+\bar{A} \cdot B \cdot \bar{C}+B \cdot \bar{B} \cdot \bar{C}+B \cdot \bar{C} \cdot C \end{aligned} $$ But from Table $11,7, \bar{A} \cdot \bar{A}=\bar{A}$ and $\bar{C} \cdot C=B \cdot \bar{B}=0$ Hence the Boolean expression becomes: $$ \begin{aligned} \bar{A} & \cdot \bar{C}+\bar{A} \cdot \bar{B} \cdot \bar{C}+\bar{A} \cdot B \cdot \bar{C} \\ &=\bar{A} \cdot \bar{C}(1+\bar{B}+B) \\ &=\bar{A} \cdot \bar{C}(1+B) \\ &=\bar{A} \cdot \bar{C} \end{aligned} $$ Thus: $\overline{(A \cdot \bar{B}+C}) \cdot(\bar{A}+\overline{B \cdot \bar{C}})=\bar{A} \cdot \bar{C}$
Please solve the third and forth question.
Shu N.
Simplified expressions: a. (x + y)(x + y') b. xyz + x'y + xyz' Express the following function as a sum of minterms and as a product of maxterms: F(A, B, C, D) = B'D + A'D + BD List the truth table of the function: F = xy + x'y' + y'z 5) Implement the Boolean function: F = xy + x'y' + y'z a. With AND, OR, and inverter gates b. With NAND and inverter gates 6) Simplify the following Boolean functions, using three-variable maps: a. F (x, y, z) = (0, 1, 5, 7) b. F (x, y, z) = (1, 2, 3, 6, 7) 7) Simplify the following Boolean functions, using four-variable maps: a. F (w, x, y, z) = Σ (1, 4, 5, 6, 12, 14, 15)
Adi S.
Recommended Textbooks
Computer Science and Information Technology
Introduction to Programming Using Python
Computer Science - An Overview
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD