dms boolean algebra tutorial

8/9/2019 DMS Boolean Algebra Tutorial

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Q9 Show that in a Boolean algebra, the modular properties

hold. That is, show that

x /\ (y V (x /\ z)) = (x /\ y) v (x /\ z) and x v (y /\ (x V z))= (x V y) /\ (x V z).

Q10 Find a minterm that equals 1 if x1= x

3= 0 and x

2= x

4= x

5= 1, and

equals 0 otherwise.

Q11 Find the sum-of-products expansion for the function

F(x, y, z) = (x + y)z.


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d) x (x + y + z)

Q17 Show that

a) x = x | x .

b) x y = (x | y) | (x | y).

c) x + y = (x | x) | (y | y).

Q18 Find the K-maps for (a) xy + x y , (b) x y + x y , and (c) x y + x y

+ x y .

Q19 Use K-maps to minimize these sum-of-products expansions.

(a) x y z+x y z+ x y z+ x y z

(b) x y z + x y z + x y z + x y z + x y z

Q20 Use K-maps to simplify these sum-of-products expansions.

(a) w x y z + w x y z + w x y z + w x y z + w x y z + w x y z +

w x y z + w x y z + w x y z

(b) w x y z + w x y z + w x y z + w x y z + w x y z + w x y z +

w x y z

Q 21


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Q23

c)


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d)

Q24


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Q25

Q26

Q27

Q28


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Q29


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Q30


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Partial Orderings

Q32 Show that the "greater than or equal" relation ( ) is a partial

ordering on the set of integers.

Q33 Show that the inclusion relation is a partial ordering on the

power set of a set S.

Q34 Let R be the relation on the set of people such that x R y if x and

yare people and x is older than y. Show that R is not a partial ordering.

NOTE : The poset (Z, < ) is totally ordered, because a < b or b < a

whenever a and b are integers.

NOTE : The set of ordered pairs of positive integers, Z+X Z+, with(a1, a2) (b1, b2) if al< b1, or if a1= b1and a2< b2(the lexicographic

ordering), is a well-ordered set.

NOTE: The set Z, with the usual < ordering, is not well-ordered because

the set of negative integers, which is a subset of Z, has no least

element.

Q 35 Is the poset (Z+, |) a lattice?

Q 36 Determine whether the posets ({ 1, 2, 3, 4, 5}, |) and ({ 1, 2, 4, 8,

16}, |) are lattices.

Q 37 Determine whether (P(S), ) is a lattice where S is a set.

Application of Lattice

The Lattice Model of Information Flow In many settings the flow of

information from one person or computer program to another is

restricted via security clearances. y.le can use a lattice model torepresent different information flow policies. For example, one common

information flow policy is the multilevel security policy used in

government and military systems. Each piece of information is assigned

to a security class, and each security class is represented by a pair

(A, C) where A is an authority level and C is a category. People and

computer programs are then allowed access to information from a


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specific restricted set of security classes. The typical authority levels

used in the U. S. government are unclassified (0), confidential (1),

secret (2), and top secret (3). Categories used in security classes are the

subsets of a set of all compartments relevant to a particular area of

interest. Each compartment represents a particular subject area. For

example, if the set of compartments is {spies, moles, double agents},

then there are eight different categories, one for each of the eight

subsets of the set of compartments, such as {spies, moles}.

We can order security classes by specifying that (A1, C1) (A2, C2) if and

only if Al< A2and C1 C2. Information is permitted to flow from

security class (A1, C1) into security class (A2, C2) if and only if (A1, C1)

(A2, C2). For example, information is permitted to flow from the

security class (secret, {spies, moles}) into the security class (top secret,

{spies, moles, double agents}), whereas information is not allowed to

flow from the security class (top secret, {spies, moles}) into either of thesecurity classes (secret, {spies, moles, double agents}) or (top secret,

{spies}).

Q 38

Which of these relations on to {0, 1, 2, 3} are partial order-

ings? Determine the properties of a partial ordering that

the others lack.

a) {(0, 0), (1,1), (2, 2), (3, 3)}

b) {(0, 0), (1, 1), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)}

c) {(0, 0), (1,1), (1, 2), (2, 2), (3, 3)}

d) {(0, 0), (1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)}

e) {(0, 0), (0,1), (0, 2), (1, 0), (1,1), (1, 2), (2, 0),

(2,2), (3, 3)}

Q 39

Which of these relations on {0,1, 2, 3} are partial order-ings? Determine the properties of a partial ordering that

the others lack.

a) {(0, 0), (2, 2), (3, 3)}

b) {(0, 0), (1, 1), (2, 0), (2, 2), (2, 3), (3, 3)}

c) {(0,0),(1, 1),(1,2),(2,2),(3, 1),(3,3)}

d) {(0, 0), (1,1), (1, 2), (1, 3), (2, 0), (2, 2), (2, 3), (3, 0), (3,3)}


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e) {(0, 0), (0,1), (0, 2), (0, 3), (1, 0), (1,1), (1, 2), (1, 3),

(2, 0), (2, 2), (3, 3)}

Q40

Is (S, R) a poset if S is the set of all people in the world

and (a, b) R, where a and b are people, if

a) a is no shorter than b?

b) a weighs more than b?

c) a = b or a is a descendant of b?

d) a and b do not have a common friend?

Q 41

Which of these are posets?

a) (Z, =) b) (Z, ) c) (Z , ) d)(Z, not divisible by)

Q 42

Which of these are posets?

a) (R, =) b) (R,


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Q 44

Let (S, R) be a poset. Show that (S, R-1

) is also a poset,

where R-1

is the inverse of R. The po set (S, R-1

) is called

the dual of (S, R).

Q45

Find the duals of these posets.

a) ({0, 1, 2}, < ) (b) (Z , ) (c) (P(Z), ) (d) (Z+, |)

Q 46

Find the lexicographic ordering of the bit strings 0, 01,11, 001, 010, 011, 0001, and 0101 based on the ordering

0 < 1.

Q 47

Draw the Hasse diagram for the "greater than or equal to"

relation on to {0, 1, 2, 3, 4, 5}.

Q 48

Draw the Hasse diagram for the "less than or equal to"relation on to {0, 2, 5, 10, 11, 15}.

Q 49

Let (S, ) be a poset. We say that an element y S covers

an element x S if x y and there is no element z S such

that x z y. The set of pairs (x, y) such that y covers x is

called the covering relation of (S, ).

Q 50 What is the covering relation of the partial ordering{(a, b) / a divides b} on {1, 2, 3, 4, 6, 12}?

Q 51 What is the covering relation of the partial ordering

{(A, B) / A B} on the power set of S, where S = {a, b, c}?


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Q 52 Show that the pair (x, y) belongs to the covering relation

of the finite poset (S, ) if and only if x is lower than y and there is an

edge joining x and y in the Hasse diagram of this poset.

Q 53

Answer these questions for the poset ({ { 1 }, {2}, {4 }, {1, 2},

{1,4}, {2,4}, {3,41, {1,3,4}, {2,3,4}} | ).

a) Find the maximal elements.

b) Find the minimal elements.

c) Is there a greatest element?

d) Is there a least element?

e) Find all upper bounds of {{2}, {4}}.

I) Find the least upper bound of {{2 }, {4 }} if it exists.g) Find all lower bounds of {{1, 3,4}, {2, 3, 4}}.

h) Find the greatest lower bound of {{ 1, 3, 4}, {2, 3, 4}}

if it exists.

dms boolean algebra tutorial

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