## Consider the universe U={x|x∈Z,1≤x≤10}U=\{x | x \in \mathbb{Z}, 1 \le x \le 10\} and the following subsets of UU

A={2,5,9} B={1,4,7,8,10} C={1,2} In the following questions we denote A′ the complement of A. i.e. A′=U−A A′ A∪B∪C B′−A A△B I can’t access my notes and finding this hard to work out. Thanks. Answer A′=U−A. A∪B∪C={x|x∈A∨x∈B∨x∈C}. B′−A=(U−B)−A. AΔB={x|(x∈A∨x∈B)∧x∉(A∩B)}={x|x∈(A∪B)∧x∉(A∩B)}. I will do 1, you can probably do the rest from here. U={1,2,3,4,5,6,7,8,9,10}. So A′=U−A=U−{2,5,9}={1,3,4,6,7,8,10}. Here are … Read more

## Prove that $X\triangle\emptyset=X$

I’m working on my proofs involving sets, though this one is not a homework problem, so if you wish to provide your own example, so be it. I am working on exercise 3.3.14 (1) in Bloch’s Proofs and Fundamentals. It asks me to prove that $X\triangle \emptyset=X$. The following is what I have so far … Read more

## Is there anything to prove in this corollary?

Show that if B is not finite and B⊂A, then A is not finite. I mean the statement is very trivial, but I’m having an issue actually writing what I would deem a good proof of this. The only idea I have is of letting x∈B and then show it is in A, but even … Read more

## Jaccard dissimilarity and the triangle inequality

Suppose that δ(A,B)=|AΔB||A∪B|, where Δ represents symmetric difference. Then how does one prove the triangle inequality, viz that δ(A,B)+δ(B,C)≥δ(A,C)? Answer If I understand you correctly, the objects you are considering are finite sets and your dissimilarity function is defined as d(A,B)=|A⊖B||A∪B|, where ⊖ denotes the symmetric difference. Consider the Venn diagram for three sets A, … Read more

## Is the Cartesian product of two uncountable sets uncountable? [duplicate]

This question already has answers here: Is the set of all pairs of real numbers uncountable? (2 answers) Closed 6 years ago. Is Cartesian product of two uncountable sets uncountable? Suppose we have a set of real numbers R, Can’t it be shown that R is uncountable by Cantor’s diagonalization method, so it follows that … Read more

## A∖(B∩C)=(A∖B)∪(A∖C)A\backslash (B\cap C) = (A\backslash B)\cup (A\backslash C); only one inclusion seems to work

I encountered the following problem: A∖(B∩C)=(A∖B)∪(A∖C). So I need to prove two things: A∖(B∩C)⊆(A∖B)∪(A∖C) (A∖B)∪(A∖C)⊆A∖(B∩C) The first one is fairly easy to me; what I don’t understand is 2. My approach was to suppose x∈A∧¬(x∈B) or x∈A∧¬(x∈C) and try to derive x∈A∧¬(x∈B)∧¬(x∈C), i.e. ∀x ((x∈A∧¬(x∈B))∨(x∈A∧¬(x∈C))→(x∈A∧¬(x∈B)∧¬(x∈C))) But this doesn’t seem possible. For all I know there are … Read more

## What is X\cap\mathcal P(X)X\cap\mathcal P(X)?

Does the powerset of X contain X as a subset, and thus X\cap \mathcal{P}(X)=X, or is X\cap \mathcal{P}(X)=\emptyset since X is a member of the \mathcal{P}(X), and not a subset? Answer Actually: A \subset X \iff A \in \wp(X). All situations can happen. If X = \{ 0,1\}, then \wp(X) = \{ \varnothing, \{0\},\{1\},\{0,1\} \}, … Read more

## A question on the generalization of Cartesian Product

In Halmos’s Book, it is written that, The notation of families is the one normally used in generalizing the concept of Cartesian product. The Cartesian product of two sets $X$ and $Y$ was defined as the set of all ordered pairs $(x,y)$ with $x$ in $X$ and $y$ in $Y$. There is a natural one-to-one … Read more

## Question about set of all functions and the power set of a set

Hi so we know the set of all functions from a set X ϕ→ {0,1} create a one to one correspondence from the power set of X to the set of all functions but we are looking at certain subsets of the set of all functions. What if we look at the set of all … Read more

## If n<ℵ∗(m)n < \aleph^*(m), then n<2mn < 2^m.

Without AC Let ℵ∗(m) be the least aleph that ≰. I need a help or hint that if n < \aleph^*(m), then n < 2^m. a \leq^* b means we can define a surjective map from b onto a. Answer HINT: If there is a surjection from A onto B, then there is an injection … Read more