Overlap in Set Theory

Question

Is this statement $$A \cap (B - C) \subseteq A - (B \cap C) $$ right? Where A, B, and C are any three events. My initial guess was that $$A \cap (B - C) = (A - B ) \cap (B - C) $$. Any proof to show this statement is true?

Answer 1

Draw a Venn Diagram with three sets and see if it is true.

Algebraically, you could write $B - C$ as $B \cap C^c$, $B$ intersected with the complement of $C$. The complement of $C$ can be interpretted as the set of all points in $A\cup B$ that are not in $C$ and then check to see if your first inequality is true. (Try to work it out.)

However, your equality is false: $$A \cap (B-C) = A \cap (B \cap C^c) = A \cap B \cap C^c$$ and $$(A- B) \cap (B-C) = (A \cap B^c)\cap (B \cap C^c) = A \cap B \cap B^c \cap C^c = \emptyset$$ because $B \cap B^c = \emptyset$. This makes sense because if you "read" what it is saying, it is saying that an element $x \in (A- B) \cap (B-C)$ is in $A$ but not $B$ and also in $B$ but not $C$, but that is not possible to be in $B$ and not in $B$.

This can all be checked using the standard "set theory" ideas of showing that a set is a subset of another by asking if an element of one set (that we want to show is the subset) is necessarily an element of the other set.

Answer 2

Is this statement $A \cap (B \smallsetminus C) \subseteq A \smallsetminus (B \cap C) $ right?

The first includes elements that are in both A and B but not in C. $A\cap (B\smallsetminus C) = A\cap B\cap C^\complement$

The second includes elements in A that are not in both B and C. $A\smallsetminus (B\cap C) = A\cap (B^\complement\cup C^\complement)$

Now $x\in A\cap B\cap C^\complement$ says $x$ is in $A$, in $B$, and not in $C$.   If it is not in $C$ we may infer it is not in both $B$ and $C$.   So $x\in A\cap(B\smallsetminus C)$ implies $x\in A\smallsetminus (B\cap C)$.

Therefore $A\cap(B\smallsetminus C) \subseteq A\smallsetminus (B\cap C)$

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Your proposed equality however, is not so true.