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

[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 !

 

[I like Serena]

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.

Let's start with the forward 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]

"

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.
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