Articles of elementary set theory

Unions and Functions on Sets

Given these conditions, I seek a proof. Let $f: A \rightarrow B$ be a function, and let $X$ and $Y$ be subsets of $A$. Prove that $f(X \cup Y) = f(X) \cup f(Y)$. I can’t seem to figure it out. It appears obvious, but materializing a proof is troubling me. What is the best method […]

What is the cardinality of the set of all functions from $\mathbb{Z} \to \mathbb{Z}$?

How can I approach this? I have to find the cardinality of the set of the functions from $\mathbb{Z} \to \mathbb{Z}$ and I have no idea on how to solve it. Can someone hint me here? The approach itself is what confuses me… how do I try to map this to something else since its […]

Defining additions in the ring of integers

Is it possible to describe all possible ways in which one can define additions in the set of integers to give it a structure of ring when the multiplication is same as the usual multiplication ? If the addition is same as the usual addition then one can easily describe all possible multiplications. What can […]

Show that 2S = S for all infinite sets

I am a little ashamed to ask such a simple question here, but how can I prove that for any infinite set, 2S (two copies of the same set) has the same cardinality as S? I can do this for the naturals and reals but do not know how to extend this to higher cardinalities.

Prove that any family of disjoint 8-signs on the plane is countable

I try to answer the following question from Basic Set Theory by Shen and Vereshchagin : 1. (a).Prove that any family of disjoint 8-signs on the plane is countable.(By an 8-sign we mean a union of two tangent circles of any size; the interior part of the circles is not included). (b)Prove a similar statement […]

Validity of empty sets

As i have read that a set is a collection of well defined objects or elements but empty set means that there is no elements in the set.We say for example “it is a set of cups,a set of pens” and like wise then what is empty set and two empty sets can be different?

The concept of ordinals

I am trying to understand this concept and have some difficulties. For example, can I say that $\alpha$ is the cardinality of $\{1,2,3,…\}=\Bbb N$? And if so, what is $\alpha +1$? I guess it is not the cardinality of $\Bbb N \cup \{\sqrt2\}$ (for example)?

Real numbers and countability

No subset of the real numbers is countable. True or false. In looking at the wikipedia article for real numbers, I’m not really clear on the answer to this. They use the words computable and countable, and I’m not sure of the difference. Also, I dont think I fully understand the term countable. The definition […]

Prove that two sets are the same

I need to prove that $(A \Delta B ) \Delta C = A \Delta (B\Delta C)$for any sets A,B,C, and $A \Delta B = (A-B)\cup(B-A)$. I tried to expand both the left hand side and the right hand side expression, and after that my expresion was composed only of sample proposition ($x \in A / […]

To characterize uncountable sets on which there exists a metric which makes the space connected

For which uncountable sets $X$ is it true that there exist a metric $d$ on $X$ such that $(X,d)$ is connected ? [ The motivation for this question is : I wanted to characterize function $f : X \to X$ such that for any topology $\tau_1,\tau_2$ on $X$ , $f:(X,\tau_1)\to (X,\tau_2)$ is continuous ; I […]