1 Binary Number Systems
1. (one’s complement) Show the operation of −10 + (−5) in 6-bit one’s comple-
ment.
2. (two’s and one’s complements) We have defined and learned the idea of two’s
and one’s complements for n-bit binary numbers. Define the corresponding
complements using an n-digit system with base 10. Show the arithmetic of −x+y
where x= 21610 and y= 6510 in the corresponding complement representations
using a 6-digit system with base 10.
3. (two’s and one’s complements) We have defined and learned the idea of two’s
and one’s complements for n-bit binary numbers. Define the corresponding
complements using an n-digit system with base 8. Show the arithmetic of −x−y
where x= 1208and y= 278in the corresponding complement representations
using a 6-digit system with base 8.
2 Boolean Algebra
1. (proof of consensus theorem) Prove the Boolean equality ab +a0c=ab +
a0c+bc.
2. (Boolean transform) Reduce the Boolean expression f(a, b, c) = ab0+ac +
a0b+a0c0.
3. (expression in sum of products) Express Boolean function
E(x, y, z) = (x+y+x0z)0(x0y0+xy 0z) in sum-of-products form.
4. (expression in product of sums) Express Boolean function
E(x, y, z) = [(x0y+x)0(x0+y)(y0+z)]0in product-of-sums form.
5. (expression in sum of products) Express Boolean function
E(a, b, c, d) = ab + (cd +bc)0+ad in sum-of-products form.
6. (expression in product of sums) Express Boolean function
E(x, y, z) = [xy 0(x0y+z)]0in product-of-sums form.
3 Recursive function
3.1 Recursive function: formulation
1. (permutation) Suppose all the permutations on the set of {1,2,3,4,5,6}are
listed in lexicographic order.
(a) What is the RANK in the list for 563241?
(b) What permutation will have the RANK 273?
2. (tower of Hanoi) Consider the Towers of Hanoi puzzle, H(6, S, E, G). Suppose
that pole Shas washers 4,1; pole Ehas washers 3,2; pole Ghas washers 6,5.
Call this the basic configuration.
(a) What is the path in the decision tree that corresponds to the basic config-
uration?
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