To determine the union of two sets [tex]\(X\)[/tex] and [tex]\(Y\)[/tex], we'll follow these steps:
1. Identify the elements of each set:
- Set [tex]\(X\)[/tex] is given as [tex]\(\{a, b, c\}\)[/tex].
- Set [tex]\(Y\)[/tex] is given as [tex]\(\{0\}\)[/tex].
2. Understand the Union operation:
The union of two sets [tex]\(X\)[/tex] and [tex]\(Y\)[/tex], denoted as [tex]\(X \cup Y\)[/tex], is a set containing all elements that are in [tex]\(X\)[/tex], or in [tex]\(Y\)[/tex], or in both. Each element is included only once regardless of how many times it appears in the original sets.
3. Combine the elements:
We now combine all distinct elements from both sets:
- From set [tex]\(X\)[/tex]: [tex]\(a, b, c\)[/tex]
- From set [tex]\(Y\)[/tex]: [tex]\(0\)[/tex]
4. Form the union:
Combine all distinct elements into one set. Since none of the elements in set [tex]\(X\)[/tex] is in set [tex]\(Y\)[/tex], all elements are included exactly once.
Hence, the union [tex]\(X \cup Y\)[/tex] is:
[tex]\[ X \cup Y = \{ a, b, c, 0 \} \][/tex]
Therefore, the union of sets [tex]\(X\)[/tex] and [tex]\(Y\)[/tex] is [tex]\(\{a, b, c, 0\}\)[/tex].
