20085eeM16_2_Homework1_Soln

20085eeM16_2_Homework1_Soln - UCLA Department of Electrical...

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UCLA Department of Electrical Engineering EEM16 Homework 1 Solutions Due October 9 th , 2008 1. Find x, y, z and v a. (43652027) 8 = (x) 16 First convert to base 2. 100 011 110 101 010 000 010 111 Now convert to base 16. 1000 1111 0101 0100 0001 0111 8F5417 x = (8F5417) 16 b. (3CB504F) 16 = (y) 8 First convert to base 2. 0011 1100 1011 0101 0000 0100 1111 Now convert to base 8. 0 011 110 010 110 101 000 001 001 111 362650117 y = (362650117) 8 c. (236) 7 + (255) 6 = (z) 9 First convert to base 10. 2*7 2 + 3*7 1 + 6 + 2*6 2 + 5*6 1 + 5 = (232) 10 Now convert to base 9. 232 / 9 = 25 R7 25 / 9 = 2 R7\ z = (277) 9 d. (v) 4 such that it is a prime number with the sum of digits in base 10 is equal to (11) 7 1*7 1 + 1 = (8) 10 1 + 7 = 8 and 17 is a prime number Convert to base 4 17 / 4 = 4 R1, 4 /4 = 1 R0, thus v = (101) 4
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2. Boolean Algebra a. Exercise 2.25 xz' + x’z = = x(xy’ + x’y)’ + x’(xy’ + x’y) By substituting for z = x(xy’)’(x’y)’ + x’xy’ + x’x’y By DeMorgan’s Law, Distributive = x(x’ + y)(x + y’) + x’y By DeMorgan’s Law, Complement = (xx’ + xy)(x + y’) + x’y By Distributive = (xyx + xyy’) + x’y By Complement, Distributive = xy + x’y By Commutative, Complement = (x + x’)y By Distributive = 1 · y By Complement b. Exercise 2.26 a + a’b + a’b’c + a’b’c’d + a’b’c’d’e = = a + a’b + a’b’c + a’b’c’(d + d’e) By Distributive = a + a’b + a’b’(c + c’(d + e)) By Distributive, Simplification = a + a’b + a’b’(c + c’d + c’e) By Distributive = a + a’b + a’b’(c + d + c’e)) By Simplification = a + a’b + a’b’(c + d + e) By Commutative, Simplification = a + a’(b + b’(c + d + e)) By Distributive = a + a’(b + c + d + e) By Simplification = a + b + c + d + e By Simplification c. Exercise 2.27 i. With the assumption that c = a * b we obtain b * c = b *(a * b) = b * (ab + a’b’) By Definition of c = b(ab + a’b’) + b’(ab + a’b’)’ By Definition of a * b = ab + a’bb’ + b’(ab)’(a’b’)’ By Distributive, DeMorgan’s Law = ab + b’(a’ + b’)(a + b) By Complement, DeMorgan’s Law
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20085eeM16_2_Homework1_Soln - UCLA Department of Electrical...

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