Let G = (V, E) be a directed graph, with source s ∈ V, sink t ∈
V, and nonnegative edge capacities {ce}. Give a polynomial-time
algorithm to decide whether G has a unique minimum s-t cut (i.e.,
an s-t of capacity strictly less than that of all other s-t
cuts).
Prove
1. Let f : A→ B and g : B → C . If g 。 f is one-to-one, then f
is one-to-one.
2. Equivalence of sets is an equivalence relation (you may use
other theorems without stating them for this one).
E ::= E + T | T
T ::= T * F | F
F ::= num | (E) Num ::= 0 | 1 | 2 | 3 | 4 | 5 | . . . . . .
.
Question: 1
a. Show the Left-most derivation for the expression: 5 * 7 + 6 * (1
+ 2).
b. Show the Right-most derivation for the expression: 5 * 7 + 6
* (1 + 2).
e) T F The larger the sample that is taken, the probability of
making a type 2 error increases.
f) T F We can never conclude that H0 is true based on taking a
random sample from from the population.
g) T F A stratified random sample is more preferred over a
simple random sample when the population can be divided into
homogeneous groups.
6. Let A = {1, 2, 3, 4} and B = {5, 6, 7}. Let f = {(1, 5),(2,
5),(3, 6),(x, y)} where x ∈ A and y ∈ B are to be determined by
you. (a) In how many ways can you pick x ∈ A and y ∈ B such that f
is not a function? (b) In how many ways can you pick x ∈ A and y ∈
B such that f : A → B...
1) a) Let k ≥ 2 and let G be a k-regular bipartite
graph. Prove that G has no cut-edge. (Hint: Use the bipartite
version of handshaking.)
b) Construct a simple, connected, nonbipartite 3-regular graph
with a cut-edge. (This shows that the condition “bipartite” really
is necessary in (a).)
2) Let F_n be a fan graph and Let a_n = τ(F_n) where τ(F_n) is
the number of spanning trees in F_n. Use deletion/contraction to
prove that a_n = 3a_n-1 - a_n-2...
Let f(x) =
(x − 1)2, g(x) = e−2x,
and h(x) = 1 + ln(1 − 2x).
(a) Find the linearizations
of f, g,
and h at
a = 0.
Lf (x) =
Lg(x) =
Lh(x) =
(b) Graph f, g,
and h and their linear approximations. For which
function is the linear approximation best? For which is it worst?
Explain.
The linear approximation appears to be the best for
the function ? f g h since it is
closer to ? f g h for a larger
domain than it is to - Select - f and g g and
h f and h . The approximation looks worst
for ? f g h since ? f g h moves
away from L faster...
Let E/F be a field extension. Let a,b be elements
elements of E and algebraic over F. Let m=[Q(a):Q] and n=[Q(b):Q].
Assume that gcd(m,n)=1. Determine the basis of Q(a,b) over Q.