Is the Number of Elements in a Set Equal to Its Power Set?

In summary: If ##|A|=1## then the power set has only one element, namely ##1##. Count the elements in each set. If ##|A|=n## then the power set has ##2^n## elements. If ##|A|=1## then the power set has only one element, namely ##1##.
  • #1
Mr Davis 97
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Homework Statement


Show that ##\bigcup \{\mathcal{P} X : X \in A \} \subseteq \mathcal{P} \bigcup A##

Homework Equations

The Attempt at a Solution


Suppose that ##c \in \bigcup \{\mathcal{P} X : X \in A \}##. Then by definition this means that ##\exists a \in A## such that ##c \in \mathcal{P} a##, or, equivalently, ##\exists a \in A## such that ##c \subseteq a##, which implies that ##c \subseteq \bigcup A## which means that ##c \in \mathcal{P} \bigcup A##.

Is this a correct proof? Also, why can't I just reverse the argument that that we have equality between sets and not just one being a subset of the other
 
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  • #2
Mr Davis 97 said:

Homework Statement


Show that ##\bigcup \{\mathcal{P} X : X \in A \} \subseteq \mathcal{P} \bigcup A##

Homework Equations

The Attempt at a Solution


Suppose that ##c \in \bigcup \{\mathcal{P} X : X \in A \}##. Then by definition this means that ##\exists a \in A## such that ##c \in \mathcal{P} a##, or, equivalently, ##\exists a \in A## such that ##c \subseteq a##, which implies that ##c \subseteq \bigcup A## which means that ##c \in \mathcal{P} \bigcup A##.

Is this a correct proof? Also, why can't I just reverse the argument that that we have equality between sets and not just one being a subset of the other

It looks ok to me. As for why not the reverse, compare the power set of ##\{1,2\}## with the union of the power sets of ##\{1\}## and ##\{2\}##.
 
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  • #3
Dick said:
It looks ok to me. As for why not the reverse, compare the power set of ##\{1,2\}## with the union of the power sets of ##\{1\}## and ##\{2\}##.
When would equality hold? It seems that it would hold iff ##|A| = 1##, but I am not sure how to prove this.
 
  • #4
Mr Davis 97 said:
When would equality hold? It seems that it would hold iff ##|A| = 1##, but I am not sure how to prove this.

Count the elements in each set. If ##|A|=n## then the power set has ##2^n## elements.
 

FAQ: Is the Number of Elements in a Set Equal to Its Power Set?

What is set theory?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects. It provides a foundation for the study of mathematical structures and is an essential tool in various areas of mathematics, such as algebra, geometry, and logic.

What is a set?

A set is a well-defined collection of distinct objects, called elements, that are considered as a single entity. The elements of a set can be anything, including numbers, letters, or even other sets.

What is the difference between a set and a subset?

A subset is a set that consists of elements from another set. In other words, all the elements of a subset are also elements of the larger set. A set can be a subset of itself, but it is not a proper subset unless it has at least one element that is not in the original set.

How do you represent sets in set theory?

Sets can be represented in various ways, including using curly braces { } and listing the elements inside, using set builder notation, or using Venn diagrams. Set builder notation is a concise and precise way to represent sets using a condition that the elements must satisfy.

What is the cardinality of a set?

The cardinality of a set is the number of elements in the set. It is denoted by |S|, where S is the set. For finite sets, the cardinality is simply the count of the elements, while for infinite sets, it can be determined using other mathematical concepts, such as the concept of infinity.

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