# union and intersection of a set meaning same thing?

I am confused about union and intersection of sets.

Is my understanding correct that if there are three sets $A,B,C$ and $x \in A$ and $x \notin B$ and $x \notin C$ then we can write $x \in A \setminus (B \cup C)$ or write $x \in A \setminus (B \cap C)$ and both expressions are the same?

Also when $x \notin A$ and $x \notin B$ meaning $x \in A^c$ and $x \in B^c$, can we write $x \in A^c \cup B^c$ or write $x \in A^c \cap B^c$ and they would both mean the same thing?

• No, but we do have $A\setminus (B\cup C)\subseteq A\setminus (B\cap C)$ – Prasun Biswas Sep 23 '17 at 20:18

Let $A = \{ 1,2,3\}$, $B=\{ 1 \}, C = \{2 \}$.

$$A \setminus (B \cup C) = \{3\}$$

$$A \setminus (B \cap C) = \{1,2,3\}$$

Try to answer the other question using an example too.

• So they are not the same when $x \in A$ and $x \notin B$ and $x \notin c$? I was looking at a proof and in this condition $A \setminus (B \cup C)$ was used and I don't understand how $A \setminus (B \cup C)$ would make sense if both aren't the same because to my understanding $x \in A$ and $x \notin B$ and $x \notin c$ means $A \setminus (B \cap C)$ – lakada Sep 23 '17 at 20:23
• @amWhy Doesn't the second also contain elements of A, that remain after discarding elements in B and C? – lakada Sep 23 '17 at 20:38
• @lakada the example has illustrated that they are not equal isn't it? – Siong Thye Goh Sep 23 '17 at 22:26
• if $x \in A , x \notin B$, and $x \notin C$, we can write it as $x \in A \cap B^c \cap C^c = A \cap (B \cup C)^c = A \setminus (B \cup C)$ by de Morgan's law. – Siong Thye Goh Sep 23 '17 at 22:28

For the second part, you need to think that if some $x$ is in the intersection of two sets then it is in the union. But if it is in the union not necessarily it is in the intersection.