# CSCI 220: Computer Architecture I Instructor: Pranava K. Jha. Simplification of Boolean Functions using a Karnaugh Map

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1 CSCI 22: Computer Architecture I Instructor: Pranava K. Jha Simplification of Boolean Functions using a Karnaugh Map Q.. Plot the following Boolean function on a Karnaugh map: f(a, b, c, d) = m(, 2, 4, 6, 7, 8,, 2, 3). Here is the basic template relating to positioning of the minterms on the map. K-map for the given function follows. c d c d cd cd a b a b ab ab of 4

2 Q. 2. Identify all prime implicants and essential prime implicants with respect to the function f(a, b, c, d) that appears on the following Karnaugh map. c d c d cd cd a b a b ab ab c d c d cd cd a b a b ab * ab * Prime implicant Essential? Distinguishing minterm {2, 6,, 4}: cd Yes abcd {,, 8, 9}: b c Yes ab c d {, 2, 8, }: b d No {, 5}: a c d No {5, 7}: a bd No {6, 7}: a bc No 2 of 4

3 Q. 3. Determine the prime implicants and essential prime implicants with respect to the function f(a, b, c, d) that appears on the following Karnaugh map. c d c d cd cd a b a b * * ab * ab * Prime implicant Essential? Distinguishing minterm {, 2, 8, }: b d Yes ab c d {8} {4, 5, 6, 7}: a b Yes a bc d {5} or a bcd {7} {, 2, 4, 6}: a d No {2, 6,, 4}: cd Yes abcd {4} 3 of 4

4 Q. 4. Identify all prime implicants and essential prime implicants with respect to the function f(a, b, c, d) that appears on the following Karnaugh map. c d c d cd cd a b a b ab ab Prime implicant Essential prime implicant? Distinguishing minterm {,, 4, 5}: a c No {, 2, 4, 6}: a d Yes a b cd {4, 5, 6, 7}: a b Yes a bcd {,, 8, 9}: b c Yes ab c d or ab c d {6, 4}: bcd Yes abcd 4 of 4

5 Q. 5. Simplify the following Boolean function using a Karnaugh map: f(a, b, c, d) = Σm(3, 4, 5, 7, 9, 3, 4, 5). c d c d cd cd a b * a b * ab * ab * Prime implicant Essential? Distinguishing minterm a cd: {3, 7} Yes a b cd {3} a bc : {4, 5} Yes a bc d {4} ac d: {9, 3} Yes ab c d {9} abc: {4, 5} Yes abcd {5} bd: {5, 7, 3, 5} No To build a simplified expression in sum-of-products form, first include the essential prime implicants. If all minterms (i.e., s on the map) are covered in this process, then stop; otherwise, additionally include some non-essential prime implicants as few as necessary so that all minterms are covered. In the present case, the essential prime implicants themselves cover all minterms. Accordingly, the simplified expression is as follows: f(a, b, c, d) = a cd + a bc + ac d + abc. Remark: One may be tempted to include the prime implicant {5, 7, 3, 5} that corresponds to the smallest product term bd. However, this prime implicant is not only non-essential but also redundant as far as final simplified expression is concerned. 5 of 4

6 Q. 6. Simplify the following Boolean function using a Karnaugh map: f(a, b, c, d) = Σm(,, 5, 7, 8,, 4, 5). c d c d cd cd a b a b ab ab Prime implicant Essential? a b c : {, } No a bd: {5, 7} No abc: {4, 5} No ab d : {8, } No b c d : {, 8} No a c d: {, 5} No bcd: {7, 5} No acd : {, 4} No Solution I: a b c + a bd + abc + ab d Solution II: b c d + a c d + bcd + acd Remark: All prime implicants are non-essential and solution is not unique. 6 of 4

7 Q. 7. A switching circuit has two control inputs C and C, two data inputs X and X, and one output Z. The circuit performs logic operations on the data inputs, as shown in the table below. Present a truth table for Z. Further, use a Karnaugh map to develop a minimum sum-of-products expression for Z in terms of C, C, X and X. Here is the truth table. C C Function performed by the circuit X X X X X + X X X C C X X Z In this case, all prime implicants are essential. The simplified expression follows: C X X + C C X X + C C X X + C C X X + C X X + C C X + C C X. 7 of 4

8 Q. 8. Use a Karnaugh map to develop a minimum product-of-sums expression for the following function: Method of attack: F(A, B, C, D) = A B + CD + ABC + A B CD + ABCD. Develop a minimum sum-of-products expression for F (A, B, C, D). Obtain the dual of the resulting expression, and complement each literal. Note that F(A, B, C, D) = m(,, 2, 3, 6,, 4, 5). Accordingly, F (A, B, C, D) = m(4, 5, 7, 8, 9,, 2, 3). A K-map for F (A, B, C, D) follows. C D C D CD CD A B A B AB AB Al prime implicants in this case are essential, and hence F (A, B, C, D) = AC + BC + A BD + AB D. Taking the dual and complementing each literal results in the following product-of-sums expression for F(A, B, C, D): (A + C)(B + C)(A + B + D )(A + B + D ). 8 of 4

9 Q. 9. A logic circuit realizes the following function: F(A, B, C, D) = A B + A CD + AC D + AB D. Assuming that A = C never occurs when B = or D =, develop a simplified expression for F in sum-of-products form. The truth table follows. A B C D f Here is the Karnaugh map. C D C D CD CD A B A B AB AB Simplified expression: B + D. 9 of 4

10 Q.. Obtain a simplified expression for the following function in sum-of-products form: f(a, b, c, d) = m(, 2, 4, 6, 7, 8,, 2, 3) + dc(5, 5) where dc stands for don t-care. Prime implicant Essential? Distinguishing minterm {, 2, 8, }: b d Yes ab cd {, 2, 4, 6}: a d No {, 4, 8, 2}: c d No {4, 5, 6, 7}: a b No {4, 5, 2, 3}: bc No {5, 7, 3, 5}: bd No Simplified expression: a b + bc + b d. of 4

11 Q. Simplify the following Boolean expression in sum-of-products form: Note that (A' + B' + D')(A + B' + C')(A' + B + D')(B + C' + D'). (A' + B' + D') = (A' + B' + C' + D')(A' + B' + C + D') = M 5 M 3 (A + B' + C') = (A + B' + C' + D')(A + B' + C' + D) = M 7 M 6 (A' + B + D') = (A' + B + C' + D')(A' + B + C + D') = M M 9 (B + C' + D') = (A' + B + C' + D')( A + B + C' + D') = M M 3 Accordingly, the given expression is representable as Π M (3, 6, 7, 9,, 3, 5) that is equivalent to Σ m (,, 2, 4, 5, 8,, 2, 4). A Karnaugh map is immediate. Simplified expression: A'C' + AD' + B'D'. of 4

12 Q. Simplify the following Boolean expression in product-of-sums form: Note that AC' + B'D + A'CD + ABCD. AC' = AB'C'D' + AB'C'D + ABC'D' + ABC'D = m 8 + m 9 + m 2 + m 3 B'D = A'B'C'D + A'B'CD + AB'C'D + AB'CD = m + m 3 + m 9 + m A'CD = A'B'CD + A'BCD = m 3 + m 7 ABCD = ABCD = m 5 Accordingly, the given expression is representable as Σ m (, 3, 7, 8, 9,, 2, 3, 5). Build the Karnaugh map as usual, but focus on s in order to derive an expression for the complement in sum-of-products form. It is clear that the complement of the given expression simplifies to A'D' + CD' + A'BC'. In order to obtain the desired simplification, get the dual of the foregoing and complement each literal: (A + D)(C' + D)(A + B' + C). 2 of 4

13 Q. Using a Karnaugh map, obtain a minimal product-of-sums expression for the following function: f(w, x, y, z) = Σm(, 2, 5, 7, 8,, 3, 5) + dc(, 4,, 4). The method of attack consists of the following steps: Obtain a simplification for the complement of the given function in sum-ofproducts form Take the dual of the resulting expression, and complement each literal. The complement of the given function is Σm(3, 6, 9, 2) + dc(, 4,, 4). The Karnaugh map follows. It is clear that the complement of the given function simplifies to: xz + x z. Taking the dual and complementing each literal leads to the desired expression: f(w, x, y, z) = (x + z)(x + z ). 3 of 4

14 Q. Design a minimal combinational network that detects the presence of any of the six illegal code groups in the 842 code by providing a logic- output. The truth table is immediate. The Karnaugh map follows w x y z f It is clear that f(w, x, y, z) = wx + wy = w(x + y). Implementation follows. 4 of 4

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