Question

let
A = { a, b, c, d , e, f, g} B = { d, e , f , g}

and C ={ a, b, c, d}

find :

(B n C)’

B’

B n C

(B U C) ‘

Answer #1

9. Let S = {a,b,c,d,e,f,g,h,i,j}.
a. is {{a}, {b, c}, {e, g}, {h, i, j}} a partition of S?
Explain.
b. is {{a, b}, {c, d}, {e, f}, {g, h}, {h, i, j}} a partition
of S? Explain. c. is {{a, b}, {c, d}, {e, f}, {g, h}, {i, j}} a
partition of S? Explain.

Let S = {a, b, c, d, e, f} with P(b) = 0.21, P(c) = 0.11, P(d) =
0.11, P(e) = 0.18, and P(f) = 0.19. Let E = {b, c, f} and F = {b,
d, e, f}. Find P(a), P(E), and P(F).

Prove the following
Let f : A → B then, for all D, E ⊆ A and for all G, H ⊆ B we
have
f-1(G ∪ H) = f-1(G) ∪
f-1(H)

Let S = {a,b,c,d,e,f,g} and let T = {1,2,3,4,5,6,7,8}.
a. How many diﬀerent functions are there from S to
T?
b. How many diﬀerent one-to-one functions are there from S to
T?
c. How many diﬀerent one-to-one functions are there from T to
S?
d. How many diﬀerent onto functions are there from T to
S?

Let the population have N=7 units, with {(unit,value)} =
{(A,-1),(B,+1),(C,-2),(D,+3),(E,-4),(F,+5),(G,-6)}. The design is
as follows: first choose A or B at random; if A then choose from
{C,D} at random, if B then choose from {E,F,G} at random. 1) Find
the first-order inclusion probabilities (note that the sample size
n is fixed at 2).
Verify (show numerically for this example) that the
Horvitz-Thompson estimator is unbiased for the population total.
(Hint: find the probability of each sample and the value...

Find the number of permutations of a, b, c, d, e, f, g and h
containing no piece ab, or cd, or acb.

Let A, B, C be sets and let f : A → B and g : f (A) → C be
one-to-one functions. Prove that their composition g ◦ f , defined
by g ◦ f (x) = g(f (x)), is also one-to-one.

Using this matrix.
A =
a
b
c
d
e
f
g
h
i
Suppose that det(A) = 5. Find the determinant of the
following matrix.
B =
a + 3g
b + 3h
c + 3i
-g
-h
-i
4d
4e
4f

Find the function whose derivative is given (i.e. solve each differential
equation). Note: E>G. Answers: A,B,D,E,F,G
a) dy/dx=6x^2. y(x)=Ax^B + C.
b) du/dt=8t(t^2 + 2). u(t)=Dt^E + Ft^G + C. ans:6

Let E, F, G be three events. Find P{E ∪ F ∪ G} function of P{E},
P{F}, P{G}, P{E ∩ F}, P{E ∩ G}, P{F ∩ G}, and P{E ∩ F ∩ G}.

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