Question

Suppose we are given 4 sets A, B, C, D such that A ⊆ B and C ⊆ D such that A and C have no elements in common. Prove or give a counterexample to the assertion that A and D also have no elements in common.​

Answer 1

Here's a counterexample: let

$B = \{1, 2, 3, 4, 5\},\quad D = \{A, B, C, D, 5\}$

We choose the subsets as follows:

$A = \{1, 5\},\quad C = \{A, B, C\}$

It is true that $A\subseteq B$ and $C\subseteq D$ and that $A\cap C=\emptyset$, but $A\cap D = \{5\}$