I have to prove or disprove this:
$$(A – C) ∩ (C – B) = \emptyset.$$
Can someone show me or guide me on how do I do this? I tried proving this using Venn diagrams, and I think it's false, because we never subtract from $B$. Is that right?
Prove or disprove this set equality.
discrete mathematicselementary-set-theory
Related Solutions
The numbers in the diagram indicate how many times the given formula counts that particular portion of the Venn Diagram. In the first picture, for example, an element in the intersection of A, B, and C will be counted 3 times. Each term of the inclusion-exclusion formula gradually refines these numbers until each portion of the Venn Diagram is counted exactly once.
Let's try necessity. Let $$c \in C \subseteq A \Delta B = (A \cup B) - (A \cap B).$$ Then, $c \in A \cup B$ and $c \not \in A \cap B$. Thus,
- every $c \in C$ also satisfies $c \in A \cup B$, hence, $C \subseteq A \cup B$;
- every $c \in C$ also satisfies $c \not \in A \cap B$, hence $C \cap A \cap B = \emptyset$.
Can you go the other way?
Best Answer
The intersection $(A-C)\cap(C-B)$ is empty,
because any element of $A-C$ is not in $C$, and any element of $C-B$ is in $C$,
so there are no elements in $A-C$ and $C-B$.