# Operations on sets

#### Yankel

##### Active member
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 !

#### Klaas van Aarsen

##### MHB Seeker
Staff member
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 $A$?

#### HallsofIvy

##### Well-known member
MHB Math Helper
"
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.