# Set theory. Is the converse true?

• omoplata
In summary, the homework statement is to prove that \cup_{x \in C} \{ 2^{x} \} \subseteq 2^{\cup C} and the homework equations are the powerset of x. 2^{x} is the powerset of x.
omoplata

## Homework Statement

Prove that $\cup_{x \in C} \{ 2^{x} \} \subseteq 2^{\cup C}$

## Homework Equations

$\cup_{x \in C} \{ 2^{x} \} = \{ A | \exists x \in C, A \subseteq 2^{x} \}$

$2^{x}$ is the powerset of $x$. i.e. $2^{x} = \{ y | y \subseteq x \}$

## The Attempt at a Solution

Suppose $A \in \cup_{x \in C} \{ 2^{x} \}$. Then,

$\exists x \in C, A \in 2^{x}$

$\exists x \in C, A \subseteq x$

$A \subseteq ( \cup C )$

$A \in 2^{\cup C}$

Therefore, $A \in \cup_{x \in C} \{ 2^{x} \} \Rightarrow A \in 2^{\cup C}$

Therefore, $\cup_{x \in C} \{ 2^{x} \} \subseteq 2^{\cup C}$

But I think there might be something wrong with my proof. Because why can't I start assuming $A \in 2^{\cup C}$ and go to $A \in \cup_{x \in C} \{ 2^{x} \}$. That means $A \in 2^{\cup C} \Rightarrow A \in \cup_{x \in C} \{ 2^{x} \}$ and therefore $2^{\cup C} \subseteq \cup_{x \in C} \{ 2^{x} \}$ also, which means $\cup_{x \in C} \{ 2^{x} \} = 2^{\cup C}$.

Is there something wrong with this proof?

omoplata said:

## Homework Statement

Prove that $\cup_{x \in C} \{ 2^{x} \} \subseteq 2^{\cup C}$

I don't like that notation. You should write it without the brackets;

$\cup_{x \in C} 2^{x} \subseteq 2^{\cup C}$

With the brackets, things become, for example

$$\{2^A\}\cup\{2^B\}=\{2^A,2^B\}$$

which is not what you want...

Anyway...

## Homework Equations

$\cup_{x \in C} \{ 2^{x} \} = \{ A | \exists x \in C, A \subseteq 2^{x} \}$

$2^{x}$ is the powerset of $x$. i.e. $2^{x} = \{ y | y \subseteq x \}$

## The Attempt at a Solution

Suppose $A \in \cup_{x \in C} \{ 2^{x} \}$. Then,

$\exists x \in C, A \in 2^{x}$

$\exists x \in C, A \subseteq x$

$A \subseteq ( \cup C )$

$A \in 2^{\cup C}$

Therefore, $A \in \cup_{x \in C} \{ 2^{x} \} \Rightarrow A \in 2^{\cup C}$

Therefore, $\cup_{x \in C} \{ 2^{x} \} \subseteq 2^{\cup C}$

But I think there might be something wrong with my proof. Because why can't I start assuming $A \in 2^{\cup C}$ and go to $A \in \cup_{x \in C} \{ 2^{x} \}$. That means $A \in 2^{\cup C} \Rightarrow A \in \cup_{x \in C} \{ 2^{x} \}$ and therefore $2^{\cup C} \subseteq \cup_{x \in C} \{ 2^{x} \}$ also, which means $\cup_{x \in C} \{ 2^{x} \} = 2^{\cup C}$.

Is there something wrong with this proof?

The proof is correct. However, you can't go backwards. The crucial step that you did, is this:$$\exists x \in C: A \subseteq x~~\Rightarrow~~A \subseteq ( \cup C )$$

This is valid (and you may want to prove this in more detail), but the converse is not (you may want to give a counterexample!).

Oh, I think I understand now. There may be a $y \in C$ such that $y \nsubseteq 2^{y}$. So $A \subseteq (\cup C) \nRightarrow A \subseteq y$. So I can't go backwards?

Sorry about the notation.

omoplata said:
There may be a $y \in C$ such that $y \nsubseteq 2^{y}$.

This makes no sense
Can you try and come up with a specific example that shows that the implication does not hold?

Oops, I meant "There may be a $y \in C$ such that $A \nsubseteq 2^{y}$."

I'll try to think of a specific example. I'll post if I can't find one.

The empty set?

omoplata said:
The empty set?

What would you take empty? Can you elaborate?

OK. I have a counterexample. Let $C = \{\{a\},\{b\}\}$ and $A = \{\{a, b\}\}$. So $A \subseteq (\cup C)$, but $\nexists x \in C : A \subseteq x$.

I think I was completely wrong in posts 3, 5 and 6.

Yes, if you meant A={a,b}. Be careful with that notation...

Oh, OK. A = {a,b}. I had actually misunderstood the definition of $\cup C$. Thanks.

## 1. What is set theory?

Set theory is a branch of mathematics that deals with the study of collections of objects, called sets, and the relationships between them.

## 2. What are the basic concepts of set theory?

The basic concepts of set theory include sets, elements, subsets, unions, intersections, and complements.

## 3. How is set theory used in mathematics?

Set theory is used as a foundational tool in mathematics, providing a rigorous framework for defining and studying mathematical objects and structures.

## 4. What is the converse of a statement in set theory?

In set theory, the converse of a statement is the opposite of the original statement, obtained by interchanging the subject and predicate.

## 5. Is the converse true in set theory?

The truth of the converse of a statement in set theory depends on the specific statement being considered. In general, the converse may or may not be true, and this can be determined by examining specific examples or using logical reasoning.

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