(related to Proposition: Set Union is Associative)

- Suppose that $A,B,C$ are sets.
- We will show that the set union is associative, i.e. $(A\cup B)\cup C=A\cup (B\cup C).$

- Let $x\in (A\cup B)\cup C.$
- By the definition of set union, we have $x\in (A\cup B)\vee x\in C.$
- Again, by the definition of set union, we have $(x\in A\vee x\in B)\vee x\in C.$
- By the associativity of disjunction, we get $x\in A\vee (x\in B\vee x\in C).$
- Again, by the definition of set union, $x\in A\vee x\in B\cup C.$
- Again, by the definition of set union, we get finally $x\in A\cup (B\cup C).$
- It follows that $(A\cup B)\cup C\subseteq A\cup (B\cup C).$

- Let $x\in A\cup (B\cup C).$
- By the definition of set union, we have $x\in A\vee x\in B\cup C.$
- Again, by the definition of set union, we have $x\in A\vee (x\in B\vee x\in C).$
- By the associativity of disjunction, we get $(x\in A\vee x\in B)\vee x\in C.$
- Again, by the definition of set union, $x\in A\cup B\vee x\in C.$
- Again, by the definition of set union, we get finally $x\in (A\cup B)\cup C.$
- It follows that $A\cup (B\cup C)\subseteq (A\cup B)\cup C.$

- It follows from the equality of sets that $A\cup (B\cup C)=(A\cup B)\cup C.$∎

**Kane, Jonathan**: "Writing Proofs in Analysis", Springer, 2016