What is Subsets: Definition and 220 Discussions

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.

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  1. Y

    MHB What is the meaning of one-to-one correspondence between subsets of S?

    What is the meaning of one-to-one correspondence between subsets of S?
  2. T

    Prove that the set of all 2-element subsets of N is denumerable.

    I am having difficulty with the following Exercise due next week. Prove that the set of all 2-element subsets of ##N## is denumerable. (Exercise 10.12 from Chartrand, Polimeni & Zhang's Mathematical Proofs: A Transition to Advanced Mathematics; 3rd ed.; pg. 262). My idea so far was...
  3. R

    MHB Why do the subsets in a partition have to be nonempty?

    "A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets." I was just wondering why the subsets must be nonempty. Is it just convention/convenient or is it because it would violate something else? Thanks!
  4. M

    Statement about topology of subsets of a metric space.

    Homework Statement . Prove that a closed subset in a metric space ##(X,d)## is the boundary of an open subset if and only if it has empty interior. The attempt at a solution. I got stuck in both implications: ##\implies## Suppose ##F## is a closed subspace with ##F=\partial S## for some...
  5. K

    MHB Let A and B be two subsets of some universal set. Prove that....

    **Let A and B be two subsets of some universal set. Prove that if $(A\cup B)^c$ = $A^c$ U $B^c$, then A = B.**Attempt: Let $x\in A$. Then $x\in A\cup B$, so $x\notin(A\cup B)^c$. By hypothesis $(A\cup B)^c=A^c\cup B^c$, so $x\notin A^c\cup B^c$. In particular, then, $x\notin B^c$, and therefore...
  6. P

    Which of the subsets of R^3 is a subspace of R^3.

    1. Which of the subsets of R3 is a subspace of R3. a) W = {(x,y,z): x + y + z = 0} b) W = {(x,y,z): x + y + z = 1} I was wondering if my answer for A is correct. Homework Equations 3. A) W = {(x,y,z): x + y + z = 0} Since, x + y + z = 0. Then, the values for all the...
  7. mnb96

    What are some examples of open subsets that are not manifolds?

    Hello, I was wondering if it is true that any open subset Ω in ℝn, to which we can associate an atlas with some coordinate charts, is always a manifold of dimension n (the same dimension of the parent space). Or alternatively, is it possible to find a subset of ℝn that is open, but it is a...
  8. O

    Can You Find the Best Constant for Sum-Free Subsets?

    A set A of non-zero integers is called sum-free if for all choices of a,b\in A, a+b is not contained in A. The Challenge: Find a constant c > 0 such that for every finite set of integers B not containing 0, there is a subset A of B such that A is sum-free and |A| ≥ c|B|, where |A| means the...
  9. S

    Induction to prove then number of subsets with elements of 3

    Homework Statement Prove that a set with n elements has \frac{n(n-1)(n-2)}{6} subsets containing exactly three elements whenever n is an integer greater than or equal to 3. Our professor wants us to use Induction. Homework Equations P(n) = \frac{n(n-1)(n-2)}{6} n \geq 3 The...
  10. Sudharaka

    MHB Distance between Compact Subsets

    Hi everyone, :) Here's a question that I couldn't find the full answer. Any ideas will be greatly appreciated. I felt that the compactness of \(F_1\) and \(F_2\) could be brought into the question using the following equivalency. However all my attempts to solve the question weren't...
  11. S

    Show that if f: A → B and E, F are subsets of A, then f(E ∪ F) = f(E)

    Homework Statement Show that if f: A → B and E, F are subsets of A, then f(E ∪ F) = f(E) ∪ f(F). Homework Equations The Attempt at a Solution My attempt: Suppose x is an element of E. Then f(x) is an element of f(E), which means f(x) is a subset of f(E). But x is in E...
  12. S

    Is a finite semigroup isomorphic to subsets of some group?

    Is any given finite semigroup isomorphic to some finite semigroup S that consists of some subsets of some finite group G under the operation of set multiplication defined in the usual way? (i.e. the product of two subsets A,B of G is the set consisting of all (and only) those elements of G that...
  13. J

    Collection of continuum subsets

    Let's define a set (collection) \mathcal{C} by the following conditions. X\in\mathcal{C} iff all following conditions hold: 1: X\subset [0,1]. 2: X is closed. 3: If x\in X and x<1, then there exists x'\in X such that x<x'. 4: For all x\in X there exists a \delta_x >0 such that...
  14. C

    Sequence of measurable subsets of [0,1] (Lebesgue measure, Measurable)

    Homework Statement Let \left\{E_{k}\right\}_{k\in N} be a sequence of measurable subsets of [0,1] satisfying m\left(E_{k}\right)=1. Then m\left(\bigcap^{\infty}_{k=1}E_{k}\right)=1. Homework Equations m denotes the Lebesgue measure. "Measurable" is short for Lebesgue-measurable. The Attempt...
  15. M

    Mathematical Reasoning and Writing - Counterexamples with subsets.

    Homework Statement Let f: A --> B be a function and let S, T \subseteq A and U, V \subseteq B. Give a counterexample to the statement: If f (S) \subseteq f (T); then S \subseteq T: Homework Equations The Attempt at a Solution PF: Assume f(S) \subseteq f(T). Let x \in...
  16. M

    Are These Subsets of R3 Subspaces?

    Homework Statement Which of the following subsets of R3 are subspaces? The set of all vectors of the form (a,b,c) where a, b, and c are... Homework Equations 1. integers 2. rational numbers The Attempt at a Solution I think neither are subspaces. IIRC, the scalar just needs to be...
  17. G

    Closed and Open Subsets of a Metric Space

    Homework Statement Let X be an infinite set. For p\in X and q\in X, d(p,q)=1 for p\neq q and d(p,q)=0 for p=q Prove that this is a metric. Find all open subsets of X with this metric. Find all closed subsets of X with this metric. Homework Equations NA The Attempt at a...
  18. G

    Bounded Subsets of a Metric Space

    Homework Statement Let X be a metric space and let E be a subset of X. Show that E is bounded if and only if there exists M>0 s.t. for all p,q in E, we have d(p,q)<M. Homework Equations Use the definition of bounded which states that a subset E of a metric space X is bounded if there exists...
  19. C

    Can a one-to-one mapping of a set into itself prove that the set is infinite?

    Homework Statement Let X be a set and let f be a one-to-one mapping of X into itself such that f[X] \subset X Then X is infinite. The Attempt at a Solution Let's assume for the sake of contradiction that X is finite and there is an f such that it maps all of the elements of X to a...
  20. S

    MHB Find the intersection value of 3 subsets

    Let a, b and c be three subsets of universe U with the following properties: n(A)= 63, n(B)=91, n(c)=44, The intersection of (A&B)= 25, The intersection of (A&C)=23, The intersection of (C&B)=21, n(A U B U C)= 139. Find the intersection of (A&B&C). I am told the answer is 10. I tried drawing...
  21. D

    Kernel subsets of transformations

    Homework Statement Let T_1,T_2:ℝ^n\rightarrowℝ^n be linear transformations. Show that \exists S:ℝ^n\rightarrowℝ^n s.t. T_1=S\circ T_2 \Longleftrightarrow kerT_2\subset kerT_1 . The Attempt at a Solution (\Longrightarrow) Let S:ℝ^n\rightarrowℝ^n be a linear transformation s.t...
  22. S

    Proving equivalence for a collection of subsets?

    Homework Statement Let f:X→Y where X and Y are sets. Prove that if {S\alpha}\alpha\inI is a collection of subsets of Y, then f-1(\cup\alpha\inIS\alpha)=\cup\alpha\inIf-1(S\alpha) Would I prove this by showing set inclusion both ways? And any hints on how to begin? Thanks.
  23. S

    Exploring the Properties of Clopen Subsets of ℝ: An Intuitive Explanation

    Prove that the only subset of ℝ with the absolute value metric that are both open and closed are ℝ and ∅. I know I'm supposed to prove by contradiction, but I'm having trouble: Suppose there exists a clopen subset A of ℝ, where A≠ℝ, A≠∅. Let [x,y] be a closed interval in ℝ, where x is in A...
  24. O

    Distance between compact subsets

    Homework Statement Let A,B be two disjoint, non-empty, compact subsets of a metric space (X,d). Show that there exists some r>0 such that d(a,b) > r for all a in A, b in B. Hint provided was: Assume the opposite, consider a sequence argument. Homework Equations N/A The Attempt...
  25. caffeinemachine

    MHB Number of p element subsets whose sum is divisible by p

    Let $S=\{ 1, 2, \ldots , 2p\}$, where $p$ is an odd prime. Find the number of $p$-element subsets of $S$ the sum of whose elements is divisible by $p$.Attempt. Let $\mathcal{K}$ be the set of all the $p$ element subsets of $S$. Let $\sigma(K)$ denote the sum of the elements of a member $K$ of...
  26. G

    Is {(x,y)∈R^2 | 2x+y≤2, x-y>4} an Open Subset of R^2?

    1. {(x,y)\in R^2 such that 2x+y<=2, x-y>4} Determine whether this subset of R^2 is open, closed or neither open nor closed. 2. I think this is an open subset but not sure how to prove it. I have rearranged the equations to give x>2, y<=-2x+1, y< x-1. I think it is open because x can get...
  27. A

    What is an empty family of subsets?

    At the beginning of a question my book is saying: "Let the universe of discourse be the set ℝ of real numbers, and let A be the empty family of subsets of ℝ.." How on Earth can an empty family contain sets which are subsets of something? A family is a set whose elements are sets. If the...
  28. J

    Determining the number of subsets of a set with a specific property

    I found an example like the problem asks, but I'm still trying to show the first part. You want the maximum number of subsets such that you can guarantee none are pairwise disjoint. I'm trying to apply my specific case to the whole problem. For a set with 3 elements, I chose all of the sets...
  29. P

    Help proving subsets of the integers

    I just started taking a foundations of math course that deals with proofs and all that good stuff and I need help on a problem that I'm stuck on: Prove: Z={3k:k\inZ}\cup{3k+1:k\inZ}\cup{3k+2:k\inZ} Z in this problem is the set of integers This is all that's given. I thought maybe I...
  30. B

    Understanding Subsets and Elements: Examining the Truth of ∅∈{0}

    Homework Statement The question asks me to determine whether the statement is true or false, the statement being .∅∈{0} Homework Equations The Attempt at a Solution I said that the statement was true, but apparently it is false. Wouldn't a set such as {1,{1}} be made up of the...
  31. S

    Proof: Number of different subsets of A is equal to 2^n?

    Homework Statement Prove that if a set a contains n elements, then the number of different subsets of A is equal to 2n. The Attempt at a Solution I know how to prove with just combinatorics, where to construct a subset, each element is either in the set or not, leading to 2n...
  32. L

    Number of Pairs of Subsets in X with Unique Elements: Sets and Subsets

    For a pair (A,B) of subsets of the set X=(1,2,...100), let A*B denote the set of all elements of X which belong to exactly one of A or B. what is number of pairs (A,B) of subsets of X such that A*B=(2,4,6,...100)? I let A =(1,2,3...50) and B=(51,52,...100) so there are 25 elememnts of...
  33. N

    How Can a Set Both Belong to and Be a Subset of P(N)?

    Hi there, here's the question I am given, i will provide the answer that I think is correct, do you mind checking it and possibly pointing out where I am wrong if I am? Give an example of a set S such that: a) S is a subset P(N) b) S belongs to P(N) c) S belongs to P(N) and |S|=5 here...
  34. A

    Subsets and sets symbol explenation (Very simple question please have a look)

    Homework Statement Hi I would like you please to look my attachement ,and explain to me the meaning of the line above M and N what's the meaning of this line?It seems to me that it acts as we should take the complementary collection of numbers . Homework Equations The Attempt at a...
  35. J

    Discrete Mathematics - Void Sets being Subsets of other Void Sets

    Homework Statement Hello. Here is the question: Determine whether or not R is some sort of order relation on the given set X. X = {∅, {∅}, {{∅}} } and R ε ⊆. I can't seem to figure out why the ordered pairs given are what they are. Homework Equations None. The Attempt at...
  36. C

    Questions about subsets and members of the set {{b},{c},d,{∅}}

    For the set G where G = {{b},{c},d,{∅}}, I believe these are correct: 1) {{c}} \subseteq G 2) {b} \subseteq G 3) {d} \subseteq G 4) d \subseteq G 5) ∅ \subseteq G 6) c \notin \varphi(G) 7) {c} \in \varphi(G) 8) {b} \subseteq \varphi(G) 9) {{d}} \notin \varphi(G) 10) ∅ \in \varphi(G)...
  37. C

    Proving σ-Algebra Generated by All Intervals in Rn Coincides with All Open Subsets of Rn

    Homework Statement Prove that the σ-algebra generated by the collection of all intervals in Rn coincides with the σ-algebra generated by the collection of all open subsets of Rn. Homework Equations A σ-algebra is a nonempty collection Σ of subsets of X (including X itself) that is closed...
  38. B

    Showing Closed Subsets of Compact Sets are Compact

    I'm trying to show that "closed subsets of compact sets are compact". I think I proved (or didn't) that every subset of a compact set is compact, which may be wrong. Here is what I've done so far, please correct me. q in A, q not in B, p in B implies p in A. Let {V_a} an open cover of A...
  39. T

    Function and subsets - inverses

    Homework Statement Let f:X \rightarrow Y and B_1, B_2 \in P(Y) where P(Y) is the power set. Prove that f^{-1}(B_1\cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2) Homework Equations The book gives this definition: Suppose f:X \rightarrow Y is a function. The function f^{-1}:P(Y) \rightarrow...
  40. Z

    Proof involving subsets of a vector space

    Homework Statement This is a problem from chapter 1.3 of Linear Algebra by F/I/S. Let W_{1} and W_{2} be subspaces of a vector space V. Prove that W_{1} \cup W_{2} is a subspace of V iff W_{1}\subseteqW_{2} or W_{2} \subseteq W_{1}. Homework Equations See attempt at solution. The...
  41. N

    Is x a Group? Testing the 4 Axioms and Multiplication Table Method

    Hey guys, I'm having an issue with a question, namely Let x be a subset of S4. Is x a group? x = {e, (123), (132), (12)(34)} I don't really understand how I can test the 4 axioms of a group and how x being a subset of S4 would help?
  42. E

    Continuous functions on dense subsets

    Hi, can someone give me pointers on this question Homework Statement Prove or provide a counterexample: If f : E -> Y is continuous on a dense subset E of a metric space X, then there is a continuous function g: X -> Y such that g(z) = f(z) for all z element of E. The Attempt at a Solution...
  43. S

    Prove all subsets of a finite set are finite

    Homework Statement Check the title Homework Equations Using the following definition of finite/infinite: A set X is infinite iff \exists f:X \rightarrow X that is injective but f(X) \not= X, i.e. f(X) \subset X. A set X is finite iff \forall f:X \stackrel{1-1}{\rightarrow} X it must follow...
  44. H

    Is this Set and Operation a Vector Space?

    question in attachment. please help!
  45. Rasalhague

    Is There a Bijection Between Subsets of Countable Sets?

    In http://www.proofwiki.org/wiki/Subset_of_Countable_Set that subsets of a countable set are countable, by enumerating the elements of the subset with the labels ni, hasn't the author implicitly assumed the conclusion, namely that the subset is countable: that its elements can be labeled by the...
  46. U

    Linear Algebra - Affine subsets, proving M = U + a is unique

    Homework Statement Let M be an affine subset of V. We then prove that if 0 ∈ M then M is a subspace. There exists a subspace U of V and a ∈ V such that M = U + a. (1) Show that the subspace U in (1) is uniquely determined by M and describe the extent to which a is determined by...
  47. T

    Real Analysis: countably infinite subsets of infinite sets proof

    Homework Statement Prove that every infinite subset contains a countably infinite subset. Homework Equations The Attempt at a Solution Right now, I'm working on a proof by cases. Let S be an infinite subset. Case 1: If S is countably infinite, because the set S is a subset...
  48. S

    MHB Questions about sets and subsets

    Hi, the question goes as follows: Given two subsets X and Y of a universal set U, prove that: (refer to picture) I'm having particular trouble on D and E, if anyone can clear this up or provide some clarification for me that would be appreciated! I know a venn diagram might not be possible...
  49. B

    Set which contains all of it's subsets

    Homework Statement Prove a set which contains all of it's subsets doesn't exist. The Attempt at a Solution Suppose such a set P exists. P := {x | x \in \wp(x)}. P \in \wp(x), so P \in P. This seems like a paradox to me, so all I have to prove is that a set can't contain itself. But how...
  50. I

    Set of finite subsets of Z+ is denumerable

    Hi I am trying to prove that P=\{X\in\mathcal{P}(\mathbb{Z^+})\;|\;X\mbox{ is finite }\} is denumerable. Now here is the strategy I am using. Let A_n=\{X\in\mathcal{P}(\mathbb{Z^+})\;|\; |X|=n\;\} So A_n are basically sets of subsets of \mathbb{Z^+} with cardinality n. So...
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