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ECE 349 Homework Assignment #2 Solutions 1. Find an expression for f = a0 b0 c0 + abc0 + a0 bc + ab0 c that uses only the exclusive-OR and complement operations. f = a(bc0 + b0 c) + a0 (b0 c0 + bc) = a(b ⊕ c) + a0 (b ⊕ c)0 f = a0 ⊕ b ⊕ c 2. The following table shows the input and output voltages for a 2-input logic gate circuit. Translate the table to a truth table using positive logic and then negative logic. What are the logic functions realized in each case? e1 e2 e0 0 0 0 0 +5 +5 +5 0 +5 +5 +5 +5 e1 0 0 1 1 e2 0 1 0 1 e0 0 1 1 1 e1 1 1 0 0 OR e2 1 0 1 0 e0 1 0 0 0 AND 3. Express each of the following as minterm expansions. (a) f (a, b, c) = a + b0 c = a0 b0 c + ab0 c0 + ab0 c + abc0 + abc Q (b) f (w, x, y) = M (0, 4, 5, 7) = x0 y 0 z + x0 yz 0 + x0 yz + xyz 0 4. Express each of the following as maxterm expansions. (a) f (a, b, c) = (a + b0 )(a0 + b + c)(b + c0 ) = (a + b0 + c0 )(a + b0 + c)(a0 + b + c)(a0 + b + c0 )(a + b + c0 ) P (b) f (x, y, z) = m(1, 3, 5, 6) = (x + y + z)(x + y 0 + z)(x0 + y + z)(x0 + y 0 + z 0 ) 5. For f (a, b, c) = (ab + c)(a0 + b0 ), find the minterm and maxterm expansions (use m and M notation). P Q f (a, b, c) = m(1, 3, 5) = M (0, 2, 4, 6, 7) 6. For the table below, express f as a minterm expansion (use m notation) and as a maxterm expansion (use M notation). a 0 0 0 0 1 1 1 1 f= b 0 0 1 1 0 0 1 1 P c f 0 0 1 1 0 X 1 1 0 0 1 X 0 1 1 0 m(1, 3, 6) + P d(2, 5) = Q M (0, 4, 7) · Q D(2, 5) 7. For the truth table of the previous problem, find the simplest expression for f . f = bc0 + a0 c 8. A bank vault has three locks with a different key for each lock. Each key is owned by a different person. In order to open the door, at least two people must insert their keys into the assigned locks. The signals A, B, and C are 1 if there is a key inserted into lock 1, 2, or 3, respectively. Write an equation for the variable Z, which is 1 iff the door should open. Z = AB + AC + BC 9. A combinational switching network has four inputs (A, B, C, D) and one output Z. The output is 1 iff three or more of the inputs are 1. Design the network using two AND gates and three OR gates. Assume that each gate has a maximum of three inputs so that it will be necessary to partially factor your logic equation before you realize it. Z = AB(C + D) + CD(A + B) = BD(A + C) + AC(B + D) = BC(A + D) + AD(B + C) 10. (Optional 2 Points) A combinational network has four inputs (a, b, c, d) and three outputs (x, y, z). The outputs represent a binary number whose value equals the number of 1’s at the input. For example, if abcd = 1011, then xyz = 011. Find the maxterm expansion for x, y, and z. Q X = M (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14) Q Y = M (0, 1, 2, 4, 8, 15) Q Z = M (0, 3, 5, 6, 9, 10, 12, 15) 11. Provide the truth table for the following circuit. A L L L L H H H H B L L H H L L H H C L H L H L H L H F L H L H L L H H A B F A C