Question

Question 4. (a) Find the minimum sum-of-products expression for each of the functions below using Karnaugh maps i) F = EA,B,C,D(2, 3, 6, 7, 12, 13, 14) (Note: the numbers in the brackets correspond to positions in the Karnaugh map where F takes the logic value 1, ie F = 1 when ABCD = 0010, 0011, 0110, etc). ii) F = []w.x,y,z(0, 2, 6, 7, 8, 14, 15) (Note: the numbers in the brackets correspond to positions in the Karnaugh map where F takes the logic value 0, ie F = 0 when WXYZ = 0000, 0010, 0110, etc). iii) F = EA,B,C,D (5, 6, 7, 12, 14, 15) + d(3, 9, 11) (Note: the numbers in the brackets correspond to positions in the Karnaugh map where F takes the logic value 1, ie F = 1 when ABCD = 0101, 0110, 0111, etc, in addition, don't cares in position 3, 9, and 11). (b) An electronic toy designed for toddlers contains flashing lights for each of the following items: the seven popular colours of rainbow; the letters A, B and C; the numbers 1, 2 and 3. What is the minimum number of bits required to encode all these items in binary and why? Aid your answer with a truth table. (c) State the Duality Principle and apply it to the following theorem to obtain its dual: X.Y + X.Z+Y.Z = X.Y + X.Z

Answer 1

ठ 1) F= EABCD (2,3,6,7,12,13,14) \АВ CD oo 10 - 01 HT ī fo FF AC + ABC + ABD Maxterms ii) F= Mwxyz (0,2,6,7,8,14,15) -> F= Ewxyz (1,3,4,5,9,10,11,12,13) --> Miuteris 10 01 yz 00 00 01 ? F = xỹ t ý z t W X Z + WXY CS Scanned with CamScanner Page No. 1

110) F = Ensco (5,6,7,12,14,15) + d(3,9,11) Colar 10 00 01 OO Х x Х 1o FEBC + ABD + ABD b) since there are 7 colours of rainbow (7), letters A,B,C (3) and numbers 1,2,3 (3) total number of possible outputs, m = 7+3+3 = 13 Hence, we require n=4 bits to represent above 13 outputs because u 27m ICS Scanned with CamScanner Page No. 2

Let the 7 colours of raiubow. be X., X2, X3, X4, X5, X6 and X,. Let the 4-bits be represented P, Q, R ands by Р R s Seleited Switch clut put 1 2 X2 3 - X3 4 xy 5 X5 O - 6 Х. 7 ха 0 8 A 9 B c 10 с 11 1 0 12 - 2 | 3 3 C) The a au | Principle states that if in Duality boolean expression all the orand AND operations are and all the l's and os are interchanged, logic it's implemente inteachanged the expression remaius plementing ему unchanged in the same. A+B.C a example CS Scanned with CamScanner A.(B+C) Page No.3

X.Y + Ž.Z + Y.za X.Y + X2 applying the duality principle (x+y).(*+z). (1+2) = (x+y).(+2) is the required deal - CS Scanned with CamScanner Page No.4