True or False: $A ∩ B = ∅ $ and $B ∩ C = ∅$, then $A ∩ C = ∅$?

Question

My reasoning by using Venn diagrams is that if the region $A∩B$ is an empty set and if the region $B∩C$ is also a null set then there is no reason for $A∩C$ to be a null set too because that region is not overlapping with the other two empty set

Therefore it should be false.

I would appreciate some feedback on my query to guide me through the thinking process.

Answer 1

If $A= C \ne \emptyset$ then it is not true.

Answer 2

To give a different extreme than @ChristianF's answer, suppose $B=\emptyset$ and $A,C$ are (almost) arbitrary.

Answer 3

When "it should be false", then "bring a counter example", say:

$$A=\{1,2,3\}\;,\;\;B=\{4\}\;,\;\;C=\{1\}$$

Answer 4

To disprove $\forall A,B,C \;(A\cap B=\phi=B\cap C\implies A\cap C=\phi)$ you must prove its negation, which is $$ (*)\quad \exists A,B,C\;(A\cap B=B\cap C=\phi\ne A\cap C).$$ To prove $(*),$ exhibiting the existence of just one such trio $A,B,C$ suffices. E.g. $A=\{0,1\}, B=\{2\}, C=\{0,3\}.$ Or $A=C\ne \phi =B.$

You do NOT want to prove that $\forall A,B,C\;(A\cap B=B\cap C=\phi \implies A\cap C\ne \phi)$ and you can't anyway. E.g. $A=B=C=\phi.$ Or $A=\{1\},B=\{2\},C=\{3\}. $

Answer 5

To give a counterexample in terms of more 'every day' objects:

$A$: Animals

$B$: Balloons

$C$: Cats