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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:

\[ x\subseteq\ y\  \]

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:

\[ P\subseteq\ Q\ \]

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