In this question, we have to find two Sets that fulfill the given condition in the question statement which are $ A\ \in\ B\ $ and also $ A\subseteq\ B\ $
The basic concept behind this question is the understanding of Sets, Subsets, and Elements in a Set.
In mathematics, a subset of a Set is a Set that has some elements in common. For example, let us suppose that $x $ is a Set having the following elements:
\[ x = \{ 0, 1 , 2 \} \]
And there is a set $ y$ which is equal to:
\[ y = \{ 0, 1, 2, 3, 4, 5 \} \]
So, by looking at the elements of both the Sets we can easily say that Set $ x$ is the subset of Set $ y$ as the elements of Set $ x$ are all present in the Set $y $ and mathematically this notation can be expressed as:
\[ x\subseteq\ y\ \]
Expert Answer
Let us suppose that the Set $ A$ has the following element(s):
\[ A = \{ \emptyset\} \]
And that Set $B $ has the following elements:
\[ B = \{ \{ \},\{1 \},\{2 \},\{3 \} \} \]
As we know that empty Set is the subset of every Set. Then we can say that the elements of Set $ A$ are also the elements of Set $ B$, which is written as:
Set $A $ belongs to Set $B $.
\[ A\ \in\ B\ \]
Therefore, we conclude that Set $A $ is a subset of Set $B $ which is expressed as:
\[ A\subseteq\ B\ \]
Numerical Results
By supposing the elements of the two Sets according to the given condition in the question having elements as follows:
Set $ A$:
\[ A = \{\} \]
And that Set $B $:
\[ B = \{ \{\},\{1\},\{2\},\{3\} \} \]
As we can see, elements of Set $ A$ are also present in Set $ B$ so we concluded that Set $A $ is a subset of Set $B $, which is expressed as:
\[ A\subseteq\ B\ \]
Example
Prove that $ P \subseteq Q$ when the Sets are:
\[ Set \space P = \{ a, b, c \} \]
\[ Set \space Q=\{ a, b, c, d, e, f, g, h\} \]
Solution:
Given that the Set $ P$ has the following element(s):
\[P = \{ a, b, c \} \]
And that Set $Q $ has the following elements:
\[Q=\{ a, b, c, d, e, f, g, h\} \]
As we can see those elements of Set $ P$ which are $a, b, c$ are also present in the Set $ Q$. Then we can say that the elements of Set $ P$ are also the elements of Set $ Q$, which is written as:
Set $P $ belongs to Set $Q $
\[ P\ \in\ Q\ \]
Therefore, we conclude that set $P $ is a subset of set $Q $ which is expressed as:
\[ P\subseteq\ Q\ \]