How to prove if $ A \subset B $ then $ A \cup C \subset B \cup C $

Question

I don't really know how to approach this question:

Let $ U $ be a universe. Use an element argument to prove the following statement.

For all sets $ A, B $ and $ C $ in $ P(U) $, if $ A \subset B $ then $ A \cup C \subset B \cup C $

My working:

Logical form: $ ( A \subset B ) \Rightarrow ( A \cup C \subset B \cup C) $

Let $ x \in ( A \subset B ) $

---

I don' really know how to do from here. Can anyone here help me understand? Thanks a lot.

Answer 1

Sometimes a figure is worth 1000 words:

Answer 2

Take an arbitrary element $x \in A \cup C$. Then $x \in A \vee x \in C$. If $x \in A$, then $x \in B$, so $x \in B \cup C$ by the properties of unions. If $x \in C$, it still follows that $x \in B \cup C$. By proving for an arbitrary element $x \in A \cup C$ that $x \in B \cup C$, we prove that $A \cup C \subseteq B \cup C$.

However, it is not necessarily true that $A \cup C \subset B \cup C$ (note the proper subset vs. subset). Suppose that $U = \mathbb{N}$, $A = \{1,2\}$, $B = \{1,2,3\}$, and $C = \{3,4,5\}$. Then clearly, $A \cup C \not \subset B \cup C$.

Note: I see you used $P(U)$, not $U$, but a similar intuition can be applied to create a counterexample for power sets.