Explanation:
To prove the associative property of the union of sets, we need to show that for three sets A, B, and C, (A ∪ B) ∪ C is equal to A ∪ (B ∪ C).
Let's start by finding (A ∪ B) ∪ C.
First, we find the union of sets A and B. A = {3, 6, 9, 12, 15} and B = {4, 8, 12, 16, 203}. Their union, (A ∪ B), will contain all the elements that are present in either A or B or both.
(A ∪ B) = {3, 4, 6, 8, 9, 12, 15, 16, 203}.
Now, we take the union of (A ∪ B) and set C. C = {5, 10, 15, 20}. The union of (A ∪ B) and C, ((A ∪ B) ∪ C), will contain all the elements that are present in either (A ∪ B) or C or both.
((A ∪ B) ∪ C) = {3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 20, 203}.
Next, let's find A ∪ (B ∪ C).
First, we find the union of sets B and C. B = {4, 8, 12, 16, 203} and C = {5, 10, 15, 20}. Their union, (B ∪ C), will contain all the elements that are present in either B or C or both.
(B ∪ C) = {4, 5, 8, 10, 12, 15, 16, 20, 203}.
Now, we take the union of set A and (B ∪ C). A = {3, 6, 9, 12, 15}. The union of A and (B ∪ C), (A ∪ (B ∪ C)), will contain all the elements that are present in either A or (B ∪ C) or both.
(A ∪ (B ∪ C)) = {3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 20, 203}.
By comparing ((A ∪ B) ∪ C) and (A ∪ (B ∪ C)), we can see that they are the same set.
Hence, we have proved the associative property of the union of sets.