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.
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...
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...
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...
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.
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...
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?
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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)...
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...
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...
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...
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...
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...
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...
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)...
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...
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 $k$.
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...
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})...
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...
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...
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 $[x-\delta, x+\delta] \subseteq I$.
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) $\left\{1+t, (1+t)^2\right\}$...
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...
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...
Let $\left(X,d\right)$ be a metric space. Let $A,B,S,T: X\to X$ be mappings satisfying
1) $T\left(X\right)\subset A(X)$ and $S\left(X\right)\subset B(X)$
2) The pairs $(S,A)$ and $(T,B)$ are weakly compatible and
3) $d\left(Sx,Ty\right)\le...
I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ...
At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ...
I need someone to confirm some...
Homework Statement
A= {a1,...am}, B= {b1,...an}. If f: A→B is a function, then f(a1) can take anyone of the n values b1,...bn. Similarly f(a2). Then there are nm such function. I understand this part.
So in my book, using this principle,
nC0 + nC1 + ... + nCn = 2n is proved.
It has taken a...
Homework Statement
If α > 1, show: ∏ (1 - \frac{z}{n^α}) converges uniformly on compact subsets of ℂ.
Homework Equations
We say that ∏ fn converges uniformly on A if
1. ∃n0 such that fn(z) ≠ 0, ∀n ≥ n0, ∀z ∈ A.
2. {∏ fn} n=n0 to n0+0, converges uniformly on A to a non-vanishing function...
Hello! (Wave)
A set is called finite if it is equinumerous with a natural number $n \in \omega$.
I want to show that the subsets of finite sets are finite sets.
That's what I have tried so far:
Let $A$ be a finite set.
Then $A \sim n$, for a natural number $n \in \omega$.
That means that...
This question asks me to prove a statement regarding linear spans. I have devised a proof but I am not sure if it is substantial enough to prove the statement.
1. Homework Statement
Let L(S) be the subspace spanned by a subset S of a linear space V. Prove that if S is a subset of T and T is a...
1. The problem statement.
consider the following sets;
C = {(x, y) ∈ R^2 : y ≥ (x + 2)^2},
D = {(x, y) ∈ R^2 : y ≥ 4x + 4}.
show that C is a subset of D.
3. Attempt at solution.
Let (x,y) be an arbitrary element of C, then
y ≥ x^2 + 4x + 4.
Rearranging the inequality gives
y - 4 ≥ x^2 +...
Homework Statement
Let Ω be the universe and A1, A2, A3, ..., An the subsets of Ω.
Prove that the number of elements of Ω that belongs to exactly p (p≤n) of the sets A1, A2, A3, ..., An is
\sum_{k=0}^{n-p}(-1)^k\binom{p+k}{k}S_{p+k}
in which
S_{0} = |\Omega|
S_{1} =...
Hi,
So I understand this problem a little, I just can't understand the ending! So saying that we have n elements, we want all the subsets consisting of r elements where r goes from 0 to n.
So we want (n choose 0) + (n choose 1) + ... + (n choose n) which is the summation of n choose r for...
Homework Statement .
Let ##X## be a set and ##\mathcal A \subset \mathcal P(X)##. Prove that there is a topology ##σ(A)## on ##X## that satisfies
(i) every element of ##A## is open for ##σ(A)##
(ii) if ##\tau## is a topology on ##X## such that every element of ##\mathcal A## is open for...
Homework Statement .
Let ##X=\{(x,y) \in \mathbb R^2 :y \geq 0\}##. If ##p=(x,y)## with ##y>0##, let
##\mathcal F_p=\{B_r(p) : 0<r<y\}##, and if ##p=(x,0)##, let ##\mathcal F_p=\{B_r(x,r) \cup \{p\}: 0<r\}##.
Then, there is a neighbourhood filter system generated on ##X## and if ##\tau=\{A...
Homework Statement
If a is both the infimum of A\subseteq \mathbb{R} and of B\subseteq \mathbb{R} then a is also the infimum of A\capB
Is this statement true or false? If true, prove it. If false, give a counterexample.
Homework Equations
The Attempt at a Solution
I think...
Define
F(A) = The set of all finite subsets of A
Seq(A) = The set of all finite sequences with elements from A
Let A be an infinite set (not necessarily countable).
I want to prove the following lines.
1. Card seq(A) \le Card(A^\omega)
2. Card A = Card seq(A) = Card F(A)
If the critical points corresponding to the global min/max of a function ##f:\mathbb{R}^2\rightarrow\mathbb{R}## lie in a subset ##A## of ##\mathbb{R}^2##, then the global min/max of ##f## in ##A## correspond to the global min/max of ##f##.
If the global min/max of ##f## lie outside of ##A##...
Hello again! :D
I am given the following exercise:
With how many ways can we choose disjoint subsets $A$ and $B$ of the set $[n]=\{1,2, \dots,n \}$,if we require that the sets $A$ and $B$ are non-empty.
Without the requirement,it would be like that:
For each element $i$,we have: $i \in A, i...