Duy-An Pham
Professor Pappas
COM 1010 HON SEC
10, September 2015
Memorandum of Introduction
Hello professor, my name is Duy-An and I am a first year student here at Wayne State. I
will be majoring in
EXAMPLE 7 How many different
Boolean functions of degree n are
there? Solution: From the product rule
for counting, it follows that there are
2n different n-tuples of 0s and 1s.
Because a Boolean func
expressions produces their duals. The
duals are x + (y 1) and (x + 0)(yz),
respectively. The dual of a Boolean
function F represented by a Boolean
expression is the function represented
by the dual of
complete, because it is impossible to
express the Boolean function F (x) = x
using these operators (see Exercise
19). We have found sets containing two
operators that are functionally
complete. Can we
(x, y, z) = xy b) F (x, y, z) = x + yz c) F (x,
y, z) = xy + (xyz) d) F (x, y, z) = x(yz + y
z) 6. Use a table to express the values of
each of these Boolean functions. a) F
(x, y, z) = z b) F (x, y,
directed graph G to find all the vertices
reachable from a vertex v G. b)
Explain how to use breadth-first search
in Gconv to find all the vertices from
which a vertex v G is reachable.
(Recall that G
is no danger of confusion, the symbol
can be deleted, just as in writing
algebraic products. Unless parentheses
are used, the rules of precedence for
Boolean operators are: first, all
complements are
to find a minimum spanning tree of
this graph. 18. Given the list of edges
and their weights of a weighted
undirected connected graph, use
Kruskals algorithm to find a minimum
spanning tree of this gr
recursively as 0, 1, x1, x2,.,xn are
Boolean expressions; if E1 and E2 are
Boolean expressions, then E1, (E1E2),
and (E1 + E2) are Boolean expressions.
Each Boolean expression represents a
Boolean fun
state of the circuit. In other words,
these circuits have no memory
capabilities. Such circuits are called
combinational circuits or gating
networks. We will construct
combinational circuits using thr
minterm x1x2x3x4x5 has the correct
set of values. By taking Boolean
sums of distinct minterms we can build
up a Boolean expression with a
specified set of values. In particular, a
Boolean sum of minte
on letter frequencies as they change as
characters are successively read, such
as adaptive Huffman coding. 8. Explain
how alpha-beta pruning can be used to
simplify the computation of the value
of a g
shown in Table 6. The identity holds
because the last two columns of the
table agree. The reader should
compare the Boolean identities in
Table 5 to the logical equivalences in
Table 6 of Section 1.3
weight, then the edge with least weight
incident to a vertex v is included in
every minimum spanning tree. 44. Find
a minimum spanning tree of each of
these graphs where the degree of each
vertex in t
Duy-An Pham
COM 1010
Informative Presentation
October 1, 2015
Informational Speech: The Black Sox Scandal
Introduction:
Attention Grabber- Youre the 8x All-Star veteran shortstop nearing the end of a
The DIA is a place that I had not personally been since I was too young to appreciate the
art enclosed within. Naturally, I was excited at the opportunity to see one of Detroits greatest
attractions a
Translate 1 0 + (0 + 1) = 0, the
equality found in Example 1, into a
logical equivalence. Solution: We
obtain a logical equivalence when we
translate each 1 into a T, each 0 into an
F, each Boolean su
built up using the basic operations of
Boolean algebra. We will provide an
algorithm for producing such
expressions. The expression that we
obtain may contain many more
operations than are necessary t
y) (x z) and x (y z) = (x y)
(x z). Showing that a complemented,
distributive lattice is a Boolean algebra
has been left as Supplementary
Exercise 39 in Chapter 9. P1: 1 CH127T Rosen-2311T MHIA017-Ro
Once it is shown that a particular
structure is a Boolean algebra, then all
results established about Boolean
algebras in general apply to this
particular structure. Boolean algebras
can be defined in
n variables are equal if and only if F
(b1, b2,., bn) = G(b1, b2,.,bn)
whenever b1, b2,.,bn belong to B. Two
different Boolean expressions that
represent the same function are called
equivalent. For i
Boolean product xyz has the value 1 if
and only if x = y = 1 and z = 0. Similarly,
the product xyz has the value 1 if and
only if x = z = 0 and y = 1. The Boolean
sum of these two products, xyz + xyz,
algebra, as can be seen from Table 6 in
Section 1.3. Similarly, the set of subsets
of a universal set U with the union and
intersection operations, the empty set
and the universal set, and the set
com
y, z) = (x + y)z = xz + yz Distributive law
= x1z + 1yz Identity law = x(y + y)z + (x
+ x)yz Unit property = xyz + xy z + xyz +
xyz Distributive law = xyz + xy z + xy z.
Idempotent law Second, we can
limited, and so on. In Exercises 3133
find a degree-constrained spanning
tree of the given graph where each
vertex has degree less than or equal to
3, or show that such a spanning tree
does not exist.
in Section 1.3. Identities in Boolean
algebra can be used to prove further
identities. We demonstrate this in
Example 10. EXAMPLE 10 Prove the
absorption law x(x + y) = x using the
other identities of
represented using the three Boolean
operators , +, and . The second
problem is: Is there a smaller set of
operators that can be used to
represent all Boolean functions? We
will answer this question by
Boolean function can be expressed as a
Boolean sum of minterms. Each
minterm is the Boolean product of
Boolean variables or their
complements. This shows that every
Boolean function can be represented
using the operator | . 18. Express each
of the Boolean functions in Exercise 3
using the operator . 19. Show that the
set of operators cfw_+, is not
functionally complete. 20. Are these
sets of opera