Write down all the possible subsets of $A = \{∅\}$ and $A=∅$

Question

I am having trouble in finding the right answer to this because I'm not sure if $A = \{\varnothing\}$ and $A=∅$ are considered the same thing or not.

Would the answer just be $∅$ for both cases or $\{\{∅\},∅\}$ for the first one and $\varnothing$ for the second question?

Thank you

Answer 1

No, in elementary set theory, $A = \varnothing$ and $B = \{ \varnothing \}$ are not the same thing.

If you write $A = \varnothing$, it means that $A$ is the empty set.

If instead you write $B = \{ \varnothing \} $ you say that $B$ is a set, which contains a single element, which is the empty set. Besides, notice how you could say that, using the two letters I’ve just employed, $A$ is an element of $B$: $A \in B$.

One other way to see how $A$ and $B$ are different is to look at the cardinality, namely the number of elements in each set. The cardinality of $A$ is zero, while the cardinality of $B$ is one.

Recall that if a set has cardinality $n$, then its powerset (the set of all possible subsets) has cardinality $2^n$.

In fact, using $\wp$ to denote the powerset, $$ \wp(A) = \wp(\varnothing) = \{ \varnothing \} = B \\ \wp(B) = \wp(\{ \varnothing \} ) = \{ \varnothing, \{\varnothing\} \} = \{ A, B\}. $$

Notice how the cardinality of $\wp(A)$ is one, and the cardinality of $\wp(2)$ is two, as expected by the aforementioned formula. In fact, $2^0 = 1$ and $2^1 = 2$.

Answer 2

You just have to strictly apply the definitions.

In set theory, two sets are considered the same thing if they have the same elements, which means that every element of set 1 is an element of set 2 and vice versa.

The set $\emptyset$ has no elements. The set $\{\emptyset\}$ has exactly one element, namely $\emptyset$.

Answer 3

Crucial to set theory is learning the precise definitions and understanding how they apply in different contexts. This question seems to center around clarifying the distinction between a set and its elements, so let's explore this in more detail.

• $A = \emptyset$ : Here we are saying that A is equal to the empty set. This is a direct statement of equality saying that $A$ is the empty set itself.

• $A$ = {$\emptyset$} : In this case, $A$ is a set that contains the empty set as its only element. Importantly, $A$ is not the empty set itself. Instead, it is a set whose only member is the empty set. This means that $A$ has exactly one element, and that element is $\emptyset$.

These two notations illustrate a fundamental concept in set theory: the difference between a set and its elements. In the first case, $A$ has no elements (it's empty), while in the second case, $A$ has one element (the empty set).

I hope this clarifies the distinction and helps you deepen your understanding of set theory. Make sure to have a read of all the various set theory definitions which will help.