For each of the following sets X and collections T of open
subsets decide whether the...
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...
Let S = {0,2,4,6} and T = {1,3,5,7}. Determine
whether each of the following sets of...
Let S = {0,2,4,6} and T = {1,3,5,7}. Determine
whether each of the following sets of ordered pairs is a function
with domain S and codomain T. If so, is it
one-to-one? Is it onto?
a. {(0,2),(2,4),(4,6),(6,0)}
b. {(6,3),(2,1),(0,3),(4,5)}
c. {(2,3),(4,7),(0,1),(6,5)}
d. {(2,1),(4,5),(6,3)}
e. {(6,1),(0,3),(4,1),(0,7),(2,5)}
14. For each of the following sets, show whether it is convex or
not.
(a) ?...
14. For each of the following sets, show whether it is convex or
not.
(a) ? = {(?1, ?2 ) ? ∈ ? 2 : ?1 + ?2 ≤ 2, ?1 ≥ 0, ?2 ≥ 0}
(b) ? = {(?1, ?2 ) ? ∈ ? 2 : ?2 − ?1 2 = 0}
Show whether the following vectors/functions form linearly
independent sets:
(a) 2 – 3x, x + 2x^2...
Show whether the following vectors/functions form linearly
independent sets:
(a) 2 – 3x, x + 2x^2 , – x^2 + x^3
(b) cos x, e^(–ix), 3 sin x
(c) (i, 1, 2), (3, i, –2), (–7+i, 1–i, 6+2i)
For each of the following sets, determine whether they are
countable or uncountable (explain your reasoning)....
For each of the following sets, determine whether they are
countable or uncountable (explain your reasoning). For countable
sets, provide some explicit counting scheme and list the first 20
elements according to your scheme. (a) The set [0, 1]R ×
[0, 1]R = {(x, y) | x, y ∈ R, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.
(b) The set [0, 1]Q × [0, 1]Q = {(x, y) |
x, y ∈ Q, 0 ≤ x ≤...
1. Write the following sets in list form. (For example, {x | x
∈N,1 ≤ x...
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)...
two vectors equal⟨1,−3,−2⟩andv=⟨−2,−1,−1⟩determineaplanein
space. Without using linear algebra and or row reduction, determine
whether the following...
two vectors equal⟨1,−3,−2⟩andv=⟨−2,−1,−1⟩determineaplanein
space. Without using linear algebra and or row reduction, determine
whether the following vectors lie in the plane formed by u and
v.
A. ⟨8, 4, 4⟩
B. ⟨1, 3, −1⟩
C. ⟨6, −4, −2⟩