Is cartesian product of two sets a subset of power set of power set of their union?

Question

Let X,Y be two sets. Is this true that "$X\times Y$ is subset of $P(P(X\cup Y)$"?

I tried to solve like this: If we choose an element of $X\times Y$, then it should be an element of

$P(P(X\cup Y)$.

Let this element be A. Assume A is an element of $X\times Y$. It means A has 2 elements, one from the set X and one from the set Y.

So, we need to show A is an element of $P(P(X\cup Y)$. It means that A is subset of $P(X\cup Y)$. Therefore, if we choose an element of A, it should be an element of $P(X\cup Y)$.

We assumed that A has 2 elements, one from the set X and one from the set Y. Let the element from set X be x and the element from set Y be y.

x and y are elements of $P(X\cup Y)$. Then, x and y are subset of $X\cup Y$. So if we choose an element of x or y, it should be an element of $X\cup Y$.

I don't know the rest of solution. Because x and y do not have to be set.

Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — José Carlos Santos, Dec 28 '21 at 11:35
@JoséCarlosSantos They may not have used MathJax properly, but in this case, with a minimum of goodwill, the question is not really impacted. I have no problems whatsoever understanding the math they write. There are other, more important issues to focus on with this post. Such as the lack of context. — Arthur, Dec 28 '21 at 11:45
You can use LaTeX/MathJax operators \times and \cup for a cross multiplication sign and for sets union. — CiaPan, Dec 28 '21 at 11:51
Your question looks like a homework. If you want the community to help you, please show your own effort – what you tried and where you stuck; what your trouble with the problem is, whether some what part is incomprehensible, etc. See also How do I ask a good question?. — CiaPan, Dec 28 '21 at 11:54
I am really sorry for bad-writing. As you guess, I am a newbie on this website. I think that it will be better with time. Thank you for your helps. — emre, Dec 28 '21 at 11:54
Just use the Edit link :) and add lacking information to your question. — CiaPan, Dec 28 '21 at 11:58
Actually, I don't know how to write the symbol of being an element of a set, the symbol of union, and the symbol of intersection. Therefore, I tried to write the problem as short as possible. — emre, Dec 28 '21 at 12:07
You need to be careful and contemplate in which way (if at all) your course considers an ordered pair $(x,y)$ a set and how this set is constructed from its components $x$ and $y$. It seems that at some point you think that $(x,y)={x,y}$, which is certainly not correct. — Hagen von Eitzen, Dec 28 '21 at 13:08

score 0 · Accepted Answer · answered Dec 28 '21 at 13:41

Your solution is not correct.

The mistake happens here:

Assume $A$ is an element of $X\times Y$. It means $A$ has $2$ elements, one from the set $X$ and one from the set $Y$.

This is not true. You should check the definitions that your textbook uses, but the most standard definitions used are as follows:

An "ordered pair" $(a,b)$ is defined as the set $\{a, \{a,b\}\}$.
The set $X\times Y$ is defined as the set of all ordered pairs $(x,y)$ where $x\in X$ and $y\in Y$.

Using the definitions I wrote above, the statement you wrote is false.

Is cartesian product of two sets a subset of power set of power set of their union?

1 Answers1