Homework #4
ECE 15a Winter
1. Write the following functions using only + (OR), • (AND) and ’(NOT) operations:
(5p) (a) f = (a ? b)(c’? d)
(5p) (b) g= a?b’? (c’? d)
Simplify the expressions for f and g as much as you can.
2. A set of gates is said to be functionally complete (or universal) if and only if every Boolean function can be realizedentirely by means of gates from this set.
(a) (8p) Prove that the f-gate defined as f(a,b,c) = b’c+a’c’+ab’ is a universal gate. (Hint: Build NOT, AND,OR gates using only the f-gates)
(b) (6p) Build a 2-input XOR gate from the f-gates defined above.
(10p) 3. Five workers: a,b,c,d and e can work to complete a certain task if the following conditions are met:
(i) Either a or b work, but not both
(ii) If b doesn’t work then c doesn’t work too
(iii) If a works then d works
(iv) b or e or both work
(v) If a doesn’t work then e works
Which workers can work together?
4. Let f(a,b,c,d) = ? m(0,2,4,5,10,11,12,13)
(2p) Show a K-Map for f(a,b,c,d).
(3p) List all prime implicants of f.
(3p) List all essential prime implicants of f.
5. A switching circuit has 2 control inputs C1 and C2, two data inputs X1 and X2, and one output Z. The circuitperforms one of the logic operations: NAND, NOR, XOR and EQU (Equivalence) on the two data inputs. The functionperformed depends on the control inputs:
C1 C2 Function
0 0 NOR
0 1 NAND
1 0 EQU
1 1 XOR
(3p) (a) Derive a truth table for Z.
(3p) (b) Use Karnaugh map to find a minimum AND-OR gate circuit to realize Z.
6. Find the minimum sum-of-products expression for each function. Underline the essential prime implicants in youranswer and tell which minterm makes each one essential.
(5p) (a) f(a,b,c,d) = ?M(2,3,6,9,13,14, 15)
(5p) (b) f(a,b,c,d)= ?m(0, 1,2,4,5,8,10,11, 13,15)
7. Find the minimum sum of products and the minimum product of sums for each function:
(3p) (a) f(a,b,c,d) = ?M(0,2,7,8,10,15) •?D(3,4,5,9)
(3p) (b) f(a,b,c,d) = ?m(1,2,10,12,13) + ?d(0,6,8,9,15)
8. f(a,b,c,d,e) = ?m(0,2 ,5, 6, 7,14,15, 16,18,19,21, 23,24,26,27,29,31)
(3p) (a) Find all prime implicants using the Karnaugh map.
(3p) (b) Find the essential prime implicants using Karnaugh map, and indicate why each one of thechosen prime implicants is essential.
9. Given F(a,b,c,d) =a’cd’+a’c’d’+bc’d’+a’bc’d+a’bc
(3p) (a) Find the maxterm expression for F. Express your answer both in decimal and algebraic form.
(3p) (b) Use Karnaugh map to find the minimum sum-of-products form for F’.
(4p) (c) Find the minimum product-of-sums for F.
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