# Math Homework

Problem 1: (15 points)

Consider the Boolean algebra (B,+, *,' , 0, 1) represented by this Hasse
diagram:

Give the tables of +, *, and '.

Problem 2: (2' points)

For any integer n > 1 let D_{n} be the set of positive divisors of n.
Define ,
and ' on

D_{n} by = lcm (a, b) (that is, least common multiple of a and b),
= gcd(a, b) (that

is, greatest common divisor of a and b), and a' = n/a. Define in D_{n} the relation
: if

a) Show that if and only if a divides b .

b) Prove that is a partial order relation in D_{n}.

c) Draw the Hasse diagrams of D_{4}, D_{6}, D_{8} and D_{10}.

d) Which of the following sets is a Boolean algebra : D_{4}, D_{6}, D_{8} and D_{10}? Explain
why.

For each Boolean algebra indicate what its 0 and 1 elements are.

Problem 3: (15 points)

Let be a Boolean algebra and let
be the Boolean function
such

that

a) Write f in disjunctive normal form and in conjunctive normal form.

b) Give the truth table of f' (the complement of f).

c) Give f' in disjunctive normal form and in conjunctive normal form.

Problem 4: (2' points)

Minimize each of the following Boolean expressions using Karnaugh maps. Show the

Karnaugh maps.

Problem 5: (15 points)

Let f and g be two Boolean functions of x, y, z, w where f(x, y, z,w) = 1 if and
only

if at least two of x , y, z, w have value 1, and g(x, y, z,w) = 1 if and only if
at most two of

x, y, z, w have value 1.

a) Give the truth tables of f and g.

b) Minimize f and g using Karnaugh maps

Problem 6: (15 points)

Let (B,+, *,' , 0, 1) be a Boolean algebra . Define
as follows:

a) Evaluate

b) Express using the operation but without using +,
*, or '.

c) Express x + y using the operation but without using +, *, or '.

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