Question

(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪ A2 ∪ ... ∪ An is countable. (Hint: Induction.)

(6) Let F be the set of all functions from R to R. Show that |F| > 2 ℵ0 . (Hint: Find an injective function from P(R) to F.)

(7) Let X = {1, 2, 3, 4}, Y = {5, 6, 7, 8}, T = {∅, {1}, {4}, {1, 4}, {1, 2, 3, 4}}, and S = {∅, {6}, {7}, {6, 7}, {5, 6, 7, 8}}. Prove that (X, T ) and (Y, S) are homeomorphic.

(8) Let X = {1, 2}, Y = {3, 4}, T = {∅, {1, 2}}, and S = {∅, {3}, {3, 4}}. Prove that (X, T ) and (Y, S) are not homeomorphic

Answer #1

Question 3. Let a1,...,an ∈R. Prove that
(a1 + a2 + ... + an)2
/n ≤ (a1)2 + (a2)2 +
... + (an)2.
Question 5. Let S ⊆R and T ⊆R be non-empty. Suppose that s ≤ t for
all s ∈ S and t ∈ T. Prove that sup(S) ≤ inf(T).
Question 6. Let S ⊆ R and T ⊆ R. Suppose that S is bounded above
and T is bounded below. Let U = {t−s|t ∈ T, s...

Let X, Y and Z be sets. Let f : X → Y and g : Y → Z functions.
(a) (3 Pts.) Show that if g ◦ f is an injective function, then f is
an injective function. (b) (2 Pts.) Find examples of sets X, Y and
Z and functions f : X → Y and g : Y → Z such that g ◦ f is
injective but g is not injective. (c) (3 Pts.) Show that...

Let a0 = 1, a1 = 2, a2 = 4, and an = an-1 + an-3 for n>=
3.
Let P(n) denote an an <= 2^n.
Prove that P(n) for n>= 0 using strong induction:
(a) (1 point) Show that P(0), P(1), and P(2) are true, which
completes
the base case.
(b) Inductive Step:
i. (1 point) What is your inductive hypothesis?
ii. (1 point) What are you trying to prove?
iii. (2 points) Complete the proof:

Let a0 = 1, a1 = 2, a2 = 4, and an = an-1 + an-3 for n>=
3.
Let P(n) denote an an <= 2^n.
Prove that P(n) for n>= 0 using strong induction:
(a) (1 point) Show that P(0), P(1), and P(2) are true, which
completes
the base case.
(b) Inductive Step:
i. (1 point) What is your inductive hypothesis?
ii. (1 point) What are you trying to prove?
iii. (2 points) Complete the proof:

Let a1 = [
7
2
-1
]
a2 =[
-1
2
3
]
a3= [
6
4
9
]
a.)determine whether a1
a2 and a3span
R3
b.) is a3 in the Span {a1,
a2}?

Answer the following brief question:
(1) Given a set X the power set P(X) is ...
(2) Let X, Y be two infinite sets. Suppose there exists an
injective map f : X → Y but no surjective map X → Y . What can one
say about the cardinalities card(X) and card(Y ) ?
(3) How many subsets of cardinality 7 are there in a set of
cardinality 10 ?
(4) How many functions are there from X =...

Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x]
(1) Prove that if then f(x) = g(x)h(x)
for some g(x), h(x) ∈ Z[x],
g(ai) + h(ai) = 0 for all i = 1, 2, ..., n
(2) Prove that f(x) is irreducible over Q

Let X and Y be sets. Prove X −(X −Y ) ⊆ X ∩Y . (Hint: Remember
that s ∈ S − T means s ∈ S and s ∈/ T . Thus, s ∈/ S − T means s ∈/
S or s ∈ T .)

Events A1,A2, and A3 form a partition of sample space
S with Pr(A1)=3/7, Pr(A2)=3/7, Pr(A3)=1/7. E is an event in S with
Pr(E|A1)=3/5, Pr(E|A2)=2/5, and Pr(E|A3)=3/5.
What is Pr(E)?
What is Pr(A2|E)?
What is Pr(E')?
What is Pr(A2'|E')?

1.- Prove the intermediate value theorem: let (X, τ) be a
connected topological space, f: X - → Y a continuous transformation
and x1, x2 ∈ X with a1 = f (x1), a2 = f (x2) ( a1 different a2).
Then for all c∈ (a1, a2) there is x∈ such that f (x) = c.
2.- Let f: X - → Y be a continuous and suprajective
transformation. Show that if X is connected, then Y too.

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