So here's the exercise for my discrete math exam that I'm trying to solve:
Study with Venn diagrams the following two laws:
- A ⊂ B ^ B ⋂ C = Ø ⇒ A ⋂ C = Ø
- C ⊂ A ⋃ B ^ A ⋂ B ⋂ C = Ø ⇒ C ⋂ A = Ø
One of the is true, the other one isn't. Which one is true and which one is not? Motivate with a Venn diagram. For the law that isnt true, exemplify with three sets A,B and C that don't fullfill the law.
End of question. So here my solution begins:
In general, I find sets to be difficult. The De Morgans laws and all other laws are explained everywhere and I understand them, but when it comes to proving stuff myself I barely understand the problem and don't know how to think or where to begin.
Take the law 1 for instance (A ⊂ B ^ B ⋂ C = Ø ⇒ A ⋂ C = Ø). It is an implication, where the prefix (first part of the implication) says: A ⊂ B ^ B ⋂ C = Ø. Now, if this should be false, since this is an implication, then the entire law must be correct since an implication with a false prefix is always true?
Anyway, the task was to draw these laws into Venn diagrams to prove them right or wrong. Here's my attempt:
I'm confused. A is a subset of B and B intersection C is an empty set. So I placed the empty set where A and B are present, but also where B intersects with C, which is in the absolute middle. But now I'm stuck. This doesn't prove anything and this is probably all wrong. Which way is the correct way to draw proof 1 (and also 2) with a venn diagram and how do I prove them true or false from that?
Edit: after googleing a little, I found that A ⊂ B would look like this in a venn diagram:
Why don't we draw it like that? Instead, the standard is to draw 3 circles even if a is a subset of b. But isnt that wrong?
Best Answer
Diagram the first premise: $A \subset B$. This means that there cannot be anything inside of $A$ that is outside of $B$. In a Venn diagram, you do this by shading the area inside of $A$ and outside of $B$ ... the shading means that that area is empty:
OK, now we add to this the second premise, which is that $B \cap C = \emptyset$. So this time, the intersection of $B$ and $C$ needs to be empty, i.e. shade that very intersection. We add this to the diagram:
This diagram represents the truth of the premises. The question is now: does this force the conclusion to be true? Well, the conclusion states that $A \cap C = \emptyset$, and if you look at the diagram, we find that indeed the intersection of $A$ and $C$ is shaded, i.e. is empty. So yes, the conclusion has to be true given the diagram, i.e. given the truth of the premises. So, this is a valid argument.
For the second problem, again start with the first premise: $C \subset A \cup B$. This means that there cannot be anything in $C$ that is outside of both $A$ and $B$, and so we shade that area:
Now for premise 2: $A \cap B \cap C = \emptyset$. So, we shade the intersection of $A$, $B$, and $C$:
Now, we ask the question: does this diagram, representing the truth of the premises, force the conclusion to be true? The conclusion says that $A \cap C = \emptyset$. So, is the intersection of $A$ and $C$ empty? Well, it could be ... but there can also be something $X$ that is in the intersection of $A$ and $C$ but outside $B$:
As such, we can quickly generate a counterexample: we need to have something that is shared by $A$ and $C$, but not $B$. ... while it should still be true that there is nothing in $C$ outside of $A$ and $B$, and nothing in the intersection of all three. OK, easy:
$A = C = \{ bananas \}$
$B = \emptyset $