# Operations on Sets: Proving A⊆B⊆C & A∪B=B∩C

• MHB
• Yankel
In summary, the conversation discusses two questions regarding operations on sets. The first question asks to prove a statement using a Venn diagram, while the second question asks to determine the characteristics of two sets given a specific operation. The summary also includes hints for solving the questions.
Yankel
Dear all,

I have two small questions regarding operations on sets.

(1) Prove that $A\subseteq B\subseteq C$ if and only if $A\cup B=B\cap C$.

(2) What can you say about sets A and B if $A\B = B$ ?

In the case of (1), I have used a Venn diagram and I understand why it is true, but struggle to prove it.

In the case of (2) I think it means that B is an empty set , am I correct ?

Thank you !

Yankel said:
Dear all,

I have two small questions regarding operations on sets.

(1) Prove that $A\subseteq B\subseteq C$ if and only if $A\cup B=B\cap C$.

(2) What can you say about sets A and B if $A\B = B$ ?

In the case of (1), I have used a Venn diagram and I understand why it is true, but struggle to prove it.

In the case of (2) I think it means that B is an empty set , am I correct ?

Thank you !

To prove (1) we need to prove both the forward direction and the backward direction.

Hint: if $A\subseteq B$ then what can we say about $A\cup B$?

For the backward direction:
Hint: suppose $a$ is an element of $A$. And we have $A\cup B=B\cap C$. Can we tell if $a$ is in $B$ or $C$?

For (2), yes, you are correct.
If $B$ contains an element, then $A\setminus B$ does not contain that element, which violates the statement.
Thus $B$ cannot contain an element and must therefore be the empty set.
Can we also say something about $A$?

"
In the case of (2) I think it means that B is an empty set , am I correct ?"

NO, you are not correct. If B is the empty set, A\B= A, not B. In order that A\B= B, A and B must both be empty.

## 1. What does it mean for A to be a subset of B?

When we say that A is a subset of B, it means that all the elements in set A are also present in set B. In other words, A is contained within B.

## 2. How can we prove that A is a subset of B?

To prove that A is a subset of B, we need to show that every element in A is also present in B. This can be done by taking an arbitrary element from A and showing that it is also present in B. If this is true for all elements in A, then A is indeed a subset of B.

## 3. What does it mean for B to be a subset of C?

Similar to the first question, when we say that B is a subset of C, it means that all the elements in set B are also present in set C. In other words, B is contained within C.

## 4. How can we prove that A∪B=B∩C?

To prove that A∪B=B∩C, we need to show that every element in A∪B is also present in B∩C and vice versa. This can be done by taking an arbitrary element from A∪B and showing that it is also present in B∩C, and then taking an arbitrary element from B∩C and showing that it is also present in A∪B. If this is true for all elements, then we can conclude that A∪B=B∩C.

## 5. Why is the concept of subset important in set operations?

The concept of subset is important in set operations because it helps us determine the relationship between different sets. It allows us to compare and analyze sets, and also helps in proving mathematical statements and theorems. In particular, the concept of subset is crucial in proving set identities, such as A∪B=B∩C, which is commonly used in various mathematical and scientific fields.

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