 # Find two sets A and B such that A ∈ B and A ⊆ B. 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\$

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\$