# Is $\emptyset \subseteq P(A)$ or $\{\emptyset\}\subseteq P(A)$, where $A$ is a set and $P(A)$ is the power set of $A$

I think my professor made an error on his answer key and I'm trying to confirm it before I bring it to his attention. He asserts only 1.) is false. I believe both 1 and 4 are false. This class is only using naïve set theory.

$$A = \{1,2,3,4\}$$ Select the statement that is false

1.) $$\{2,3\} \subseteq P(A)$$ 2.) $$\{2,3\} \in P(A)$$ 3.) $$\emptyset \in P(A)$$ 4.) $$\emptyset \subseteq P(A)$$

• $$P(A)$$ is the set of all the subsets of $$A$$
• $$P(A) = P(A) = \{\emptyset,\{1\},\{2\},\{3\},\{4\},\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\},\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\},\{1,2,3,4\}\}$$
1. FALSE
• $$2$$ is not an element of $$P(A)$$
• $$3$$ is not an element of $$P(A)$$
• $$\{2,3\}$$ cannot be a subset of $$P(A)$$
• $$\{{2,3}\}$$ would be a subset of $$P(A)$$
2. TRUE
• The element $$\{2,3\}$$ can be found in the set $$P(A)$$
3. TRUE
• The element $$\emptyset$$ can be found in the set $$P(A)$$
4. FALSE
• Both operands of the subset operator requires a set. $$\emptyset$$ is the empty set were as $$\{\emptyset\}$$ is an element that is the empty set. Therefore $$\emptyset$$ is not a subset of $$P(A)$$.

## 3 Answers

#4 is true. The empty set is a subset of every set.

It happens that the empty set us also an element of this set.

So both $$\varnothing \subseteq \mathscr{P}(A)$$ and $$\varnothing \in \mathscr{P}(A)$$ are true.

• Okay, so the empty set is implicit. Does this mean that 1st element I listed in P(A) can express the following statement: {{}} ⊆ P(A)? – Shawn Armstrong Jan 7 at 2:27
• You’ve made several errors. You have correctly included the empty set, but you have omitted all subsets with 3 or 4 elements, and note that $\{2\}$ and $\{2,2\}$ are the same set containing the single element $2$. – MPW Jan 7 at 2:35

You are wrong: both $$\varnothing$$ and $$\{\varnothing\}$$ are subsets of $$\mathcal P(A)$$. $$\varnothing$$ is a subset of any set $$X$$, because the condition of all its elements being elements of $$X$$ is vacuously true; $$\{\varnothing\}$$ is a subset of $$\mathcal P(A)$$ because $$\varnothing\in\mathcal P(A)$$, seeing as $$\varnothing\subseteq A$$.

$$\{\varnothing\}$$ is a set, specifically the set containing exactly the element $$\varnothing$$. Its existence is implied by the axiom of pair and the axiom of empty set together.

The empty set is a subset of every set, so it is also true that $$\emptyset \subset P(A)$$.

• @Sumanta: the power set is always non-empty, so it is appropriate to use the symbol ‘$\subset$‘. – Pietro Paparella Jan 7 at 3:10