In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.
For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of more than two sets is called disjoint if any two distinct sets of the collection are disjoint.
I came up with two different forms of the sample space S, but I am not sure if they mean the same thing or the first one could mean something different. H stands for heads showing up and T stands for tails showing up.
$$ S = \{ \{i,j\}: i \in \{H,T\}, j \in \{H,T\} \} $$
$$ S = \{ (i,j) : i...
So, my approach and solution are as follows:
$$
[x * y] = \{ z \in M : z \sim (x * y) \}
$$
Since we know that ##a * b \sim a^{\prime} * b^{\prime}## we can rewrite ##z## as ## x^{\prime} * y^{\prime} ##. Plugging this in yields:
$$
[x * y] = \{ x^{\prime}, y^{\prime} \in M : x^{\prime} *...
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...
Suppose f1,f2... is a sequence of functions from a set X to R. This is the set T={x in X: f1(x),... has a limit in R}. I am confused about what is the meaning of the condition in the set. Is the limit a function or a number value? Why?
A set is nothing more than a collection. To determine whether or not an object belongs to the set , we test it against one or more conditions. If it satisfies these conditions then it belongs to the set, otherwise it doesn't.
The geometric point of view of sets- a set can be viewed as being...
I have a hypothesis of which I wonder if it's sound. Perhaps you guys can advise me:
Suppose ##x_n\Rightarrow a_n## (logical implication) for some set X and set A. I think we have to assume a bijection.
Then, if ##a_m = False##, ##x_m## should be ##False##, right?
So, in case of a bijection...
I tried to name the shaded area of a Venn diagram using numbers to isolate the regions. And I found that there are several ways to get the same region.
Can the set notations simplfy
Homework Statement
Suppose ##R## and ##S## are relations on a set ##A##.
If ##R## and ##S## are transitive, is ##R \cup S## transitive? Why?
Homework Equations
The Attempt at a Solution
Suppose that ##a## is an arbitrarily but particularly picked element of ##R \cup S##, then
$$a \in R \...
In my book, the definition of surjection is given as follows:
Let A and B be sets and f:A->B. The function f is said to be onto if, for each b ϵB, there is at least one a ϵ A for which f(a)=b. In other words, f is onto if R(f)=B. A function which is onto is also called a surjection or a...
Homework Statement
I'm in need of some help to be able to determine the supremum and infimum of the following sets:A = \left\{ {mn\over 1+ m+n} \mid m, n \in \mathbb N \right\}B = \left\{ {mn\over 4m^2+m+n^2} \mid m, n \in \mathbb N \right\}C = \left\{ {m\over \vert m\vert +n} \mid m \in...
Homework Statement
This excresice is supposed to help you understand the basic operations of sets, later used in probability. I am given the following phrases and have to write them in using mathematics.
Given three events A, B and C, which belong to sample space S, calculate the following...
So we just recently did accumulation points in my maths class for chemists. I understood everything that was taught but ever since I was trying to find a reasonable explanation if the sequence an = (-1)n has 2 accumulation points (-1,1) or if it doesn't have any at all. I mean it's clear that...
Homework Statement
[/B]
1. Suppose {v1, . . . , vk} is a linearly independent set of vectors in Rn and suppose A is an m × n matrix such that Nul A = {0}.
(a) Prove that {Av1, . . . , Avk} is linearly independent.
(b) Suppose that {v1, . . . , vk} is actually a basis for Rn. Under what...
I've started Book of Proof, the first chapter of which is an intro to sets.
Q.1 Is there any particular way to approach these kinds of problems, other than using intuition / trial & error? I tend to have some difficulty in working out the best way to express the general term of a sequence, for...
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...
Homework Statement
Attached is the problem
Homework Equations
The Attempt at a Solution
So I have to show that each side is a subset of the other side
Assume x∈ A ∪ (∩Bi)
so x∈A or x∈∩Bi
case 1 x∈ ∩ Bi
so x∈ (B1∩B2∩B3...∩Bn)
which implies x∈B1 and x∈B2 ... and x∈Bn
so x∈B1∪A and x∈B2∪A...
Suppose that A\B is disjoint from C and x∈ A . Prove that if x ∈ C then x ∈ B .
So I know that A\B∩C = ∅ which means A\B and C don't share any elements.
But I don't necessarily understand how to prove this. I heard I could use a contrapositive to solve it, but how do I set it up. Which is P...
Homework Statement
given set C = {(x,y)|x,y are integers, x^2 + |y| <= 2}
Suppose they are uniformly distributed and we pick a point completely at random, thus p(x,y)= 1/11
Homework Equations
Listing it all out,
R(X) = {-1,-2,0,1,2} = R(y)
The Attempt at a Solution
My problem is that when I...
Homework Statement
Let ε = { (-∞,a] : a∈ℝ } be the collection of all intervals of the form (-∞,a] = {x∈ℝ : x≤a} for some a∈ℝ.
Is ε closed under countable unions?
Homework Equations
Potentially De Morgan's laws?
The Attempt at a Solution
Hi everyone,
Thanks in advance for looking at my...
Homework Statement
Let ##A, B, C## be sets with ##A \subseteq B##. Show ##(A-B)\cup C=(A\cup C)-(B\cup C)##
Homework Equations
None.
The Attempt at a Solution
So, generally, one shows two sets to be equal by showing that each is a proper subset of the other. I started with the LHS...
Homework Statement
Let A and B be two events such that
P(A) = 0.4, P(B) = 0.7, P(A∪B) = 0.9
Find P((A^c) - B)
2. Homework Equations
I can't think of any relevant equations except maybe the Inclusion Exclusion property.
P(A∪B) = P(A) + P(B) - P(A∩B)
This leads us to another thing
P(A∩B^c)...
I was some youtube videos and i got sucked into this channel called "numberphile". They were talking about infinite sets. In particular the set that is the sum of all natural numbers. Through some creative algebra they demonstrate the proof. Somehow the set that is equal to the sum of all...
Hi,
So I was just going through my copy of The Emperor's New Mind, and I'm having a little difficulty accepting Godel's theorem , at least the way Penrose has presented it.
If I'm not wrong, the theorem asserts that there exist certain mathematical statements within a formal axiomatic system...
Homework Statement
If I am given ##n(A)## and ##n(B)## for two sets A and B, and also provided with ##n(A\cap B)^2##. We are supposed to find ##n((AXB) \cap (BXA))##.
Homework Equations
My teacher said that the formula for ##n((AXB) \cap (BXA)) = n(A \cap B)^2##. I am not sure how do you get to...
Hello all,
I probably should have posted this in a math forum but I don't know of any. Can anyone recommend a book/books on elementary number theory with exercises? My math background is not very strong with very little knowledge of set theory so it should be understood by me. We're covering...
Homework Statement
Prove that set of all onto mappings of A->A is closed under composition of mappings:
Homework Equations
Definition of onto and closure on sets.
The Attempt at a Solution
Say, ##f## and ##g## are onto mappings from A to A.
Now, say I have a set S(A) = {all onto mappings of A...
Hi there!
I would like to understand the basic concepts of how mathematics is builded up from sets of elements to anything else (i.e: an equation)
For example here is a formula: 2-1
So are there different sets of elements like
A=(2) B=(-) C=(1) ?
And maybe there is an operation with sets...