# Set theory: Problem 3

### Hints

Remember the outline for a subset inclusion proof

• Pick an element x in $$(A-B) \cup (B-C)$$
• Show that x lives in $$(A \cup B) - (A \cap B \cap C)$$.

Then spell out what it means for x to be in $$(A-B) \cup (B-C)$$, i.e. produce a logical expression where the propositions are basic set inclusions like $$x \in A$$.

Do a similar translation with your goal. To show x lives in a set difference $$X - Y$$, you need to show two things: x is a member of X and x is not a member of Y. It's often best to attack these two parts separately.

Your given information involves the OR of two possibilities. The standard approach to this situation is to use proof by cases.

Be organized and careful. Use parentheses if there's any danger of misinterpreting your own logical expressions.