# 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.

View More On Wikipedia.org
1. ### I Proof about pre-images of functions

The problem reads: ##f:M \rightarrow N##, and ##L \subseteq M## and ##P \subseteq N##. Then prove that ##L \subseteq f^{-1}(f(L))## and ##f(f^{-1}(P)) \subseteq P##. My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very...
2. ### I Is the solution to this problem as trivial as I think?

The problem goes as follows: Let ##M, N## be sets and ##f : M \rightarrow N##. Further let ##L \subseteq M## and ##P \subseteq N##. Then show that ##L \subseteq f^{-1}(f(L))## and ##f^{-1}(f(P)) \subseteq P##. Obviously, I would simply use the definition of a functions inverse to obtain...
3. ### Finding all subsets of a list of positive integers using backtracking

The following Python 3 code is provided as the solution to this problem (https://leetcode.com/problems/subsets/solution/) that asks to find all subsets of a list of integers. For example, for the list below the output is [[], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]. I am not familiar...
4. ### Is it possible to make graphs of subsets of Rational Numbers in Mathem

Is it possible to make subsets of rational numbers in Mathematica using the plot command, or any other command? Ie., say I want to graph the set of rational numbers from 0 to 1.
5. ### MHB How is lexicographic order used in ranking and unranking subsets?

Hey! :giggle: I am looking at the following codes: It is lexicographic order related to ranking and unranking. Here is also an example: There is also the Gray code: with the repective examples: I haven't really understood the ranking and the unranking. So we have a set and...
6. ### How many subsets are in {∅} and {0}?

For ##{∅}##, I've come to the conclusion that there is only one subset because it has the empty set and itself as subsets. In this case, there are the same thing. For ##{0}##, there should be two subsets; the empty set and the set itself. Am I right?
7. ### MHB Subsets of permutation group: Properties

Hey! 😊 Let $G$ be a permutation group of a set $X\neq \emptyset$ and let $x,y\in X$. We define: \begin{align*}&G_x:=\{g\in G\mid g(x)=x\} \\ &G_{x\rightarrow y}:=\{g\in G\mid g(x)=y\} \\ &B:=\{y\in X\mid \exists g\in G: g(x)=y\}\end{align*} Show the following: $G_x$ is a subgroup of $G$. The...
8. ### I Closed Subsets in a Toplogical space ....

I am reading Sasho Kalajdzievski's book: "An Illustrated Introduction to Topology and Homotopy" and am currently focused on Chapter 3: Topological Spaces: Definitions and Examples ... ... I need some help in order to fully understand Kalajdzievski's definition of a closed set in a...
9. ### MHB Countably Dense Subsets in a Metric Space .... Stromberg, Lemma 3.44 .... ....

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Lemma 3.44 on page 105 ... ... Lemma 3.44 and its proof read as follows: In the above...
10. ### MHB Understanding Stromberg's Theorem 3.18 & its Proof

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.18 on pages 98-99 ... ... Theorem 3.18 and its proof read as follows: In the...
11. ### MHB Open Subsets in a Metric Space .... Stromberg, Theorem 3.6 ... ....

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.6 on page 94 ... ... Theorem 3.6 and its proof read as follows: In the above...
12. ### MHB Compact Subsets of R .... Sohrab, Proposition 4.1.8 .... ....

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of R and Continuity ... ... I need help in order to fully understand the proof of Proposition 4.1.8 ...Proposition 4.1.8 and its proof read as follows:In the above proof by...
13. ### I Compact Subsets of R .... Sohrab, Proposition 4.1.8 .... ....

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of R and Continuity ... ... I need help in order to fully understand the proof of Proposition 4.1.8 ...Proposition 4.1.8 and its proof read as follows: In the above proof by...
14. ### I Compact Subsets of R .... Sohrab, Proposition 4.1.1 (Lindelof) ....

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of ##\mathbb{R}## and Continuity ... ... I need help in order to fully understand the proof of Proposition 4.1.1...Proposition 4.1.1, some preliminary notes and its proof read...
15. ### MHB Compact Subsets of R .... Sohrab, Proposition 4.1.1 (Lindelof)

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of \mathbb{R} and Continuity ... ... I need help in order to fully understand the proof of Proposition 4.1.1...Proposition 4.1.1, some preliminary notes and its proof read as...
16. ### How many subsets of size three or less are in a n-object set

Homework Statement (a) How many ways can at most three people out of a selection of ##n## applicants be selected for a job? (b) How many subsets of size at most three are there in a set of size ##n##? (c) How many ways can a given subset of size three or fewer be chosen for the job? Homework...
17. ### MHB Proper Subsets and Relations of Sets

Q1: Write all proper subsets of S = {1, 2, 3, 4 }. Q2: Let S = {1,2,5,6 } Define a relation R on S of at least four order pairs, as (a,b)  R iff a*b is even (i.e. a multiply by b is even)...
18. ### MHB Exploring Basis Subsets in a 5-Element Vector Space

Hey! :o Let $V$ be a vector space with with a 5-element basis $B=\{b_1, \ldots , b_5\}$ and let $v_1:=b_1+b_2$, $v_2:=b_2+b_4$ and $\displaystyle{v_3:=\sum_{i=1}^5(-1)^ib_i}$. I want to determine all subsets of $B\cup \{v_1, v_2, v_3\}$ that form a basis of $V$. Are the desired subsets the...
19. ### MHB Understanding Proper Subsets of Ordinals in Searcoid's Theorem 1.4.4 - Peter

I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ... I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ... I have another question regarding the proof of Theorem 1.4.4 ... Theorem 1.4.4 reads as follows: In the above...
20. ### I Proper Subsets of Ordinals .... .... Another Question .... ....

I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ... I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ... I have another question regarding the proof of Theorem 1.4.4 ... Theorem 1.4.4 reads as follows: In the above...
21. ### MHB Proper Subsets of Ordinals .... .... Searcoid, Theorem 1.4.4 .... ....

I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ... I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ... I need some help in fully understanding Theorem 1.4.4 ... Theorem 1.4.4 reads as follows: In the above proof...
22. ### I Proper Subsets of Ordinals .... .... Searcoid, Theorem 1.4.4 .

I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ... I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ... I need some help in fully understanding Theorem 1.4.4 ... Theorem 1.4.4 reads as follows: In the above proof...
23. ### Question about a function of sets

Let a function ##f:X \to X## be defined. Let A and B be sets such that ##A \subseteq X## and ##B \subseteq X##. Then which of the following are correct ? a) ##f(A \cup B) = f(A) \cup f(B)## b) ##f(A \cap B) = f(A) \cap f(B)## c) ##f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)## d) ##f^{-1}(A \cap...
24. ### Proving S is a Subset of T in R³

Homework Statement Show that S ⊆ T, where S and T are both subsets of R^3. Homework Equations S = {(1, 2, 1), (1, 1, 2)}, T ={(x,y,3x−y): x,y∈R} The Attempt at a Solution I considered finding if S is a spanning set for T but I'm aware that this is perhaps not relevant. If I find {α(1, 2, 1)...
25. ### MHB Open Subsets of R^n .... D&K Lemma 1.2.5

I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Lemma 1.2.5 (ii) ... Duistermaat and Kolk"s statement and proof of Lemma 1.2.5 reads as follows: My question...
26. ### MHB Find the maximal number of subsets, k.

Let $A_1, A_2, … , A_k$ be distinct subsets of $\left \{ 1,2,...,2018 \right \}$, such that for each $1 \leq i < j \leq k$ the intersection $A_i \cap A_j$ forms an arithmetic progression. Find the maximal value of $k$.
27. ### B Subsets of Rational Numbers and Well-Ordered Sets

This isn't original or anything, but I was thinking about how would one go about formalizing (in a general sense) an informal wikipedia picture such as this: https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Omega-exp-omega-labeled.svg/487px-Omega-exp-omega-labeled.svg.png For example...
28. ### I Sets, Subsets, Possible Relations

Given a set, there are subsets and possible relations between those arbitrary subsets. For a given example set, the possible relation between the subsets of the example set will narrow down to the "true" possible relations between those subsets. a) {1} Number of Subsets: ##2^1 = 2## (∅, {1})...
29. ### Are the following subsets open in the standard topology?

Homework Statement Determine whether the following subsets are open in the standard topology: a) ##(0,1)## b) ##[0,1)## c) ##(0,\infty)## d) ##\{x \in (0,1) : \forall n \in \mathbb{Z}^{+}## ##, x \not= \frac{1}{n}\} ## Homework EquationsThe Attempt at a Solution a) ##(0,1)## is open because...
30. M

### Combinatorics: looking for an alternative solution

Homework Statement Show that every subset with 6 elements of {1,2,3,4, ..., 9} contains 2 elements with sum 10. I solved this (solution below) but I want to do this easier using the pidgeon hole principle. Homework Equations Pidgeon hole principle Combinatorics The Attempt at a Solution...
31. ### MHB Proving Subsets of Intervals in $\mathbb{R}$

Let $I \subseteq \mathbb{R}$ be an interval. Prove that 1. If $x, y \in I$ and $x \le y$ then $[x,y] \subseteq I$. 2. If $I$ is an open interval, and if $x \in I$, then there is some $\delta > 0$ such that $[x-\delta, x+\delta] \subseteq I$.
32. ### MHB Subspace spanned by subsets of polynomials

In the linear space of all real polynomials $p(t)$, describe the subspace spanned by each of the following subsets of polynomials and determine the dimension of this subspace. (a) \left\{1,t^2,t^4\right\}, (b) \left\{t,t^3,t^4\right\}, (c) \left\{t,t^2\right\}, (d) $\left\{1+t, (1+t)^2\right\}$...
33. ### Clarification: proof that perfect subsets of R^k uncountable

From Baby Rudin "Thm: Let P be a non-empty, perfect subset of R^k. Then P is uncountable. Pf: Since P has limit points, P must be infinite. Suppose P is countable, list the point of P {x1 ...xn }. Construct a sequence of nbhds. as follows. Let V1 be any nbhd of x1 . Suppose Vn has been...
34. ### Inner Joins between Subsets of the Same Table.

Hi, All, I am trying to figure out the syntax for doing joins between subsets of the same table. I have: Employee ( EmpId PK , EmpFirst, EmpLast, EmpMid, DateHired, SSN, DateBirth, Gender, PhoneNum, ReportsTo) And I want to find , for each employee, the person they report to. So I am...
35. ### MHB Exploring Limits and Infinite Subsets of $\Bbb{N}$

İn a finite set, can we take limit to $\infty$ ? Also, can you give an example related to infinite subset of $\Bbb{N}$ ?