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:

\[ P\subseteq\ Q\ \]