Question

For each of the following, give a specific example of
sets that satisfy the stated conditions.

(a) A and B are infinite and |B −A| = 3.

(b) A is infinite, U is infinite, and Ac is infinite. (c) A is
infinite, U is infinite, and Ac is finite.

Answer #1

For each of the following, give an example of a function g and a
function f that satisfy the stated conditions. Or state that such
an example cannot exist. Be sure to clearly state the domain and
codomain for each function.
(a)The function g is a surjection, but the function fog is not a
surjection.
(b) The function g is not an injection, but the function fog is
an injection.
(c)The function g is an injection, but the function fog...

1) Give an example for each of the following:
a) A random experiment with finite space of elementary
events.
b) A random experiment with infinite countable space of
elementary events.
c) A random experiment with continuous space of elementary
events.

Which of the following sets are finite? countably infinite?
uncountable? Give reasons for your answers for each of the
following:
(a) {1\n :n ∈ Z\{0}};
(b)R\N;
(c){x ∈ N:|x−7|>|x|};
(d)2Z×3Z
Please answer questions in clear hand-writing and show me the
full process, thank you (Sometimes I get the answer which was
difficult to read).

For each of the following sets X and collections T of open
subsets decide whether the pair X, T satisfies the axioms of a
topological space. If it does, determine the connected components
of X. If it is not a topological space then exhibit one axiom that
fails.
(a) X = {1, 2, 3, 4} and T = {∅, {1}, {1, 2}, {2, 3}, {1, 2, 3},
{1, 2, 3, 4}}.
(b) X = {1, 2, 3, 4} and T...

Give two examples each of sets that a) are denumerable b) are
not denumerable c) are finite. Briefly explain why each set (above)
belongs to each classification.

For each set of conditions below, give an example of a predicate
P(n) deﬁned on N that satisfy those conditions (and justify your
example), or explain why such a predicate cannot exist.
(a) P(n) is True for n ≤ 5 and n = 8; False for all other
natural numbers.
(b) P(1) is False, and (∀k ≥ 1)(P(k) ⇒ P(k + 1)) is True.
(c) P(1) and P(2) are True, but [(∀k ≥ 3)(P(k) ⇒ P(k + 1))] is
False....

T
or F and explanation
1.There are 5! possible sets of five specific names.
2. A ∪ A ′ = ∅
3. If A and B are finite sets, then n(A ∪ B) = n(A) +
n(B)
4. C(10, 9) = C(10, 1)
5. If A and B are finite sets, then n(A ∪ B) = n(A) + n(B) −
n(A ∩ B)

If A and B are both uncountably infinite sets then A - B could
be? Select one of the following options:
a) Uncountably infinite
b) Finite
c) Countably infinite
Explain the solution.

1. Write the following sets in list form. (For example, {x | x
∈N,1 ≤ x < 6} would be {1,2,3,4,5}.) (a) {a | a ∈Z,a2 ≤ 1}. (b)
{b2 | b ∈Z,−2 ≤ b ≤ 2} (c) {c | c2 −4c−5 = 0}. (d) {d | d ∈R,d2
< 0}.
2. Let S be the set {1,2,{1,3},{2}}. Answer true or false: (a) 1
∈ S. (b) {2}⊆ S. (c) 3 ∈ S. (d) {1,3}∈ S. (e) {1,2}∈ S (f)...

For each problem below, either give an example of a function
satisfying the give conditions, or explain why no such function
exists.
(a) An injective function f:{1,2,3,4,5}→{1,2,3,4}
(b) A surjective function f:{1,2,3,4,5}→{1,2,3,4}
(c) A bijection f:N→E, where E is the set of all positive even
integers
(d) A function f:N→E that is surjective but not injective
(e) A function f:N→E that is injective but not surjective

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