In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B.
The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.
Homework Statement
Suppose that E,F are sets of vectors in V with E \subseteq F. Prove that if E is linearly dependent, then so is F.
The Attempt at a SolutionRead post #2. This proof, I think, was incorrect.
If we suppose that E is linearly dependent, then we know that there exists...
Homework Statement
Let Tx and Ty be topologies on X and Y, respectively. Is T = { A × B : A\inTx, B\inTy } a topology on X × Y?
The attempt at a solution
I know that in order to prove T is a topology on X × Y I need to prove:
i. (∅, ∅)\inT and (X × Y)\inT
ii. T is closed under...
The following two definitions are taken directly from Rudin's Principles of Mathematical Analysis.
(1) OPEN SUBSET DEFINITION: If G is an open subset of some metric space X, then G \subset X and for any p \in G we can find some r_{p} > 0 such that the conditions d(p,q) < r_p, q \in X implies...
Hello, I was wondering this, what is the cardinality of the set of all finite subsets of the real interval [0,1]
It somehow confuses me because the interval is nonnumerable (cardinality of the continuos \mathfrak{c}), while the subsets are less than numerable (finite). It is clear that it has...
I would very much like some help to the following problem.
Homework Statement
Using mathematical induction, prove that a finite set A of n elements has n(n-1)/2 subsets of two elements.
The Attempt at a Solution
* Base step n=2: 2(2-1)/2= 1 subset of two elements.
* Inductive step: assuming...
Hi! I have this two related questions:
(1) I was thinking that \mathbb{Q} as a subset of \mathbb{R} is a closed set (all its points are boundary points).
But when I think of \mathbb{Q} not like a subset, but like a topological space (with the inherited subspace topology), are all it's...
Homework Statement
The Attempt at a Solution
I've got through this question up to the last bit.
I've got B(0,1) = \{0\} and B(0,2) = \{y\in\mathbb{R} : -1<y<1 \} (i.e. the open interval (-1,1).)
How do I show that every subset of \mathbb{R} is open (A \subseteq X is open if it...
Homework Statement
Given a set of n elements one after another (1,2,...n) Find the minimum n for which there are about 1000 subsets such that every two subsets will have at least 2 elements not in common
Homework Equations
The Attempt at a Solution
I did this problem for 1 element...
Homework Statement
I'm just trying to find liminf and limsup for:
Homework Equations
E_n = (n, n+2)
The Attempt at a Solution
Since every subset occurs a finite number of times, would I say that limsup is the empty set? Is being bounded by nothing the same thing as being bounded by...
This is not a homework question, just a question that popped into my head over the weekend.
My apologies if this is silly, but would you say that the symmetric group S4 is a subset of S5? My friends and I are having a debate about this. One argument by analogy is that we consider the set...
1. What are all the open subsets of the subspace Z of R.
2. Homework Equations : def of openness
3. I think the solution is all the subsets of Z, but I can't see how, for example you can say the subset of Z: {1} has a B(1,r) with r>0 is contained in {1}.
Thanks for any help.
Homework Statement
The following is all the information needed:
Homework Equations
There are, of course, all the basic rules of logic and set identities to be considered.
The Attempt at a Solution
Not really sure how to attempt this one, to be honest. I know that (A ⊆ B) can...
Hi, All:
This is a post from another site that was interesting but was not
answered:
can I reasonably
> argue that three planes in 3-space are not likely
> to intersect at a point using the fact that
>t GL(3,R);
> the subset of invertible 3x3-matrices has measure 0
> in...
Homework Statement
G = {e, a, a^2, a^3, b, y, D (delta), T (theta)} Where e=(1), a=(1, 2, 3, 4), a^2 = (1, 3)(2, 4), a^3 = (1, 4, 3, 2), b = (1, 4)(2, 3), y = (2, 4), D = (1, 2)(3, 4), T = (1, 3)
Find the subsets.
Homework Equations
I know that the order of G is 8. So, my subsets...
Hi, this is my first time posting here, and I am trying to prove the following proofs and I do not know how to start:
Suppose R and S are relations on set A
1. If R is reflexive and R is a subset of S, then S is reflexive.
2. If R is symmetric and R is a subset of S, then S is symmetric.
3...
Hi, All:
Let X be a metric space and let A be a compact subset of X, B a closed subset of X. I am trying to show this implies that d(A,B)=0.
Please critique my proof:
First, we define d(A,B) as inf{d(a,b): a in A, b in B}. We then show that compactness of A forces the existence of a in A...
Homework Statement
Prove that if A=B if and only if A \subseteq B and B \subseteq A
The Attempt at a Solution
If A is a subset of B then all the elements of A are in B . And if B is a subset of A then all the elements of B are in A . There fore there is a one-to-one correspondence...
I was reading an introductory chapter on probability related to sample spaces. It had a mention that for uncountably infinite sets, ie. in sets in which 1 to 1 mapping of its elements with positive integers is not possible, the number of subsets is not 2^n.
I certainly find this very...
This may be a dumb question but let's say i have the set of integers \mathbb{z}
can I say that \frac{\pi}{\pi} or (sin(x))^2+(cos(x))^2
is a subset of the integers?
Homework Statement
I'm stuck on how to start this. The Hammin metric is define:
http://s1038.photobucket.com/albums/a467/kanye_brown/?action=view¤t=hamming_metric.jpg
and I'm asked to:
http://i1038.photobucket.com/albums/a467/kanye_brown/analysis_1.jpg?t=1306280360
a) prove...
Let R be a set of real numbers derived from rational numbers and R* be a set consisting of all ordered pairs of the form (x,0) where x is contained in R.
Then R* can be identified with R.
I'd like to ask you two questions.
1. Definition of definite integral of complex valued function of...
Homework Statement
Identify the compact subsets of \mathbb{R} with topology \tau:= \{ \emptyset , \mathbb{R}\} \cup \{ (-\infty , \alpha) | \alpha \in \mathbb{R}\} .
just need help on how would you actually go about finding it. I usually just find it by thinking about it.
The Attempt at a...
Homework Statement
1. Find an uncountable number of subsets of metric spaces \left(\mathbb{R}^{n},d_{p}\right) and \left(\mathbb{C}^{n},d_{p}\right) that are neither open nor closed.
2. If 1\leq p<q , then the unit ball in \left(\mathbb{R}^{n},d_{p}\right) is contained in the unit ball in...
Homework Statement
Suppose {V1, V2, ..., Vp} form a linearly independent set of vectors. Show that any subset of this collection of vectors is also linearly independent. Is it necessarily true that is the vectors are dependent, that any subset is also dependent?
Homework Equations
The...
Homework Statement
I reduced another problem to the following problem:
If I is an interval and A is a subset of I, then A is either an interval, a set of discreet points, a union of the two.
Homework Equations
The Attempt at a Solution
Is this trivial?
Homework Statement
Suppose A_{\lambda}, \lambda in L, represents a partition of the nonempty set A. Define R on A by xRy <=> there is a subset A{\lambda} such that x is in A{\lambda} and y is in A{\lambda}. Prove that R is an equivalence relation on A and that the equivalence classes of R are...
Alright this is a pretty simple question. So if A is a infinite subset of the reals, does that mean that its length is infinite (eg. the set of all numbers from 0 to infinite) or could it also be just a set with an infinite number of terms (eg. the set of all real numbers between 1 and 2)?
I am currently covering Set Theory from the book, A Transition to Abstract Mathematics (Douglas Smith) and have a question about subsets and an implication. The statement reads as follows:
If B is a subset of A, then {B} is an element of the Power Set A.
I choose this to be true. By...
Homework Statement
How can I find the base and dim of U here?, V = P3; U = {p in P3 : p'(0) = p(1)}...
Homework Equations
The Attempt at a Solution
now I've proven it is a subspace and that it is closed under addition and scalar multiplication...but how can I find the base and...
f: A-->B is a function. A,B are sets.
Let A1, A2 be contained in/equal to A.
f(A1 intersection A2) is contained in OR equal to f(A1) intersection with f(A2). Show that the equality holds if f is an injection.
I know how to prove that it is contained, but not the equal/injection part...
i'm not sure if I'm posting this in the right place, so forgive me if I'm wrong! in my linear algebra revision i found that I'm struggling with one of the questions:
Let S and T be subsets of a vector space, V. Which of the following statements are true? Give a proof or a counterexample.
a)...
Homework Statement
Let f:R-->R be a function. Define A={(x,y) \in R2: y<f(x)}, B={(x,y) \in R2: y>f(x)}, i.e A is the subset of R2 under the graph of f and B is the subset above the graph of f. Show that if A and B are open subsets of R2, then f is continuous
Homework Equations
N/A...
Homework Statement
Can you please explain in as much detail as possible the two following problems?
1.Each set is a function from set A to set B. a.What is the largest subset of the real numbers that can be set A, the domain of the given function? b. If set A=set B, is the function onto...
Homework Statement
Let F be the vector space (over R) of all functions f : R−R. Determine whether or not the following subsets of F are subspaces of F:
Homework Equations
1. S1 = {f e F|f(−3) = 0 and f(10) = 0};
2. S2 = {f e F|f(−3) = 0 or f(10) = 0}.
The Attempt at a Solution
I...
Homework Statement
Prove that a bounded subset of R is totally bounded.
Homework Equations
The Attempt at a Solution
Fix E > 0. Let A be subset of R, x be contained in A, and B(E/2, a) where E/2 is the radius of the ball and a is the center.
Assume that B(E/2, a) is closed...
Homework Statement
Let T be a family of finite subsets of the natural numbers N = {1, 2, 3,...} such that if A and B are any members of T, then the intersection of A and B is nonempty.
(a) Must N contain a finite subset F such that the intersection of A, B and F is nonempty for any sets A...
Homework Statement
Prove that if A is a subset of B then int(A) is a subset of int(B).
int(A) = interior of A
int(B) = interior of B
The Attempt at a Solution
Take some y E int(a) , this implies that B(r,y) is a subset of A.
Given that A is a subset of B, we know that B(r,y) is a subset...
Homework Statement
List all the subsets of set B.
B = {4,8,12}
Homework Equations
A={5,10,15,20} C={4,8,12,16} D={2,4,6,8,10} E={4,12}
The Attempt at a Solution
I know that the equation for finding the number of subsets is 2n, but I don't exactly understand how I'm supposed to...
Show that if f: A-->B, and A(1), A(2) are both subsets of A, then
Show that if f: A-->B, and A(1), A(2) are both subsets of A, then
f(A1 ∩ A2) C(is the subset of) f(A1) ∩ f(A2).
Give an example of a situation where the inclusion is strict.
Homework Statement
Show that the set of all finite subsets of N is a countable set.
The Attempt at a Solution
At first I thought this was really easy. I had A = {B1, B2, B3, ... }, where Bn is some finite subset of N. Since any B is finite and therefore countable, and since a union of...
Homework Statement
If f is continuous and f(x)=0 for all x in a dense subset of the real numbers, then f(x)=0 for all x \in \mathbb{R}.
Homework Equations
N/A
The Attempt at a Solution
Does this solution work? And if it does, can it be improved in some way?
Proof: From the...
Homework Statement
Prove that any well-ordered subset (under the natural order) of the real numbers is countable.
Homework Equations
None.
The Attempt at a Solution
My attempt thus far has been to prove by contradiction. I didn't see a very clear way to get from well-ordered subset...
Homework Statement
The problem is attached. Please help me out in understanding this problem. This is not a HW question, just for my own understanding...
Homework Equations
The Attempt at a Solution
Homework Statement
Let their be a set A, and let B be the set: {A, {A}} (the set containing the elements A and the
set that contains element A)
As you know, A is an element of B and {A} is also an element of B.
Also, {A} is a subset of B and {{A}} is also a subset of B.
However...
Dear Friends,
I have a question and would be pleased if you help me by suggesting a paper or book to study.
Let A={1,2,...,n}. We consider all the subsets with k elements. How many of these sets have a sum of r ?
e.g.
for n=6, k=3, r=10
{1,3,6}
{1,4,5}
{2,3,5}
Hense the...
Homework Statement
Identify the compact subsets of \mathbb{Q} \cap [0,1] with the relative topology from \mathbb{R}.
Homework Equations
The Attempt at a Solution
Is it all finite subsets of \mathbb{Q} \cap [0,1]? The relative topology contains single rational points in [0,1]...
Homework Statement
View S^3 as the unit sphere in C^2. Now,
1. What are the path connected components of the subset of S^3 described by the equation x^3 + y^6 = 0, where the x and y refer to the coordinates (in C)?
2. Is it true that the similar subset x^2 + y^5 = 0 is homeomorphic to...
Homework Statement
This is a question about mathematical probability, using the sigma-algebra, measure and probability space approach.
Define A(t) = {all outcomes, w, in the sample space such that Y(w) < or = t}
where Y is a random variable and t is any real number.
Fix a real number...
We want to find a basis for W and W_perpendicular for W=span({(i,0,1)}) =Span({w1}) in C^3
a vector x =(a,b,c) in W_perp satisfies <w1,x> = 0 => ai + c = 0 => c=-ai
Thus a vector x in W_perp is x = (a,b,-ai)
So an orthonormal basis in W would be simply w1/norm(w1) ...but the norm(w1)=0 (i^2 +...