# Proof

(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. $(A\cup B)\cup C=A\cup (B\cup C).$

### Part 1: $(A\cup B)\cup C\subseteq A\cup (B\cup C).$

• Let $x\in (A\cup B)\cup C.$
• By the definition of set union, we have $x\in (A\cup B)\vee x\in C.$
• 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 $(A\cup B)\cup C\subseteq A\cup (B\cup C).$

### Part 2: $B\subseteq A$

• 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 $A\cup (B\cup C)\subseteq (A\cup B)\cup C.$

### Conclusion

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

Github: ### References

#### Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016