1
$\begingroup$

If $A, B, C$ are sets prove that $(A\setminus B)\setminus C\subseteq A\setminus(B\setminus C)$. Find a description of when it is that we have equality, and give an example where the inclusion is strict.

I'm not sure how to do this one I know that to prove that two sets $X$ and $Y$ are equal, we need to show that if $x$ is any element of $X$, then $x\in Y$, and that if $x$ is any element of $Y$, then $x\in X$: the first of these means that $X\subseteq Y$ and the second that $Y\subseteq X$, so the two together establish that $X=Y$. I'm not sure how to use this for a question and how to finish it.

$\endgroup$
1
$\begingroup$

Hint:

$(A\setminus B)\setminus C=A\setminus(B\cup C)$

$A\setminus V =A \setminus U$ iff $V\cap U\cap A=U\cap A=V\cap A$.

$\endgroup$
  • $\begingroup$ What does this strict means? $\endgroup$ – Nikola Sep 13 '13 at 1:28
  • $\begingroup$ U is strictly contained in V if U is contained in V and there exists $x\in V$ such that $x\notin U$. $\endgroup$ – Pocho la pantera Sep 13 '13 at 1:47
1
$\begingroup$

We have $(A \setminus B) \setminus C = A \setminus (B \cup C)$, and $A \setminus (B \cup C) \subseteq A \setminus (B \setminus C)$ because $B \setminus C \subseteq B \cup C$. The inclusion is strict whenever $A \cap C$ is nonempty.

$\endgroup$
0
$\begingroup$

A different approach would be using DeMorgans law and basic set operations

$$ (A \setminus B) \setminus C = A \cap B^c \cap C^c $$

and

$$ A \setminus (B \setminus C) = A \setminus (B \cap C^c) = A \cap (B^c \cup C) = (A \cap B^c) \cup (A \cap C) $$

Now it should be clear that

$$ (A \setminus B) \setminus C = A \cap B^c \cap C^c \subseteq A \cap B^c \subseteq (A \cap B^c) \cup (A \cap C) = A \setminus (B \setminus C) $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.