Certainly! Let's solve the problem step-by-step.
We are given two sets [tex]\(D\)[/tex] and [tex]\(G\)[/tex]:
[tex]\[ D = \{-2, -1, 2, 3\} \][/tex]
[tex]\[ G = \{0, 3, 5, 8\} \][/tex]
### 1. Find the Intersection of [tex]\(D\)[/tex] and [tex]\(G\)[/tex]
The intersection of two sets is the set of elements that are common to both sets. To find the intersection, we look for elements that appear in both [tex]\(D\)[/tex] and [tex]\(G\)[/tex].
Looking at the sets:
- [tex]\(D\)[/tex] contains the elements [tex]\(-2, -1, 2, 3\)[/tex]
- [tex]\(G\)[/tex] contains the elements [tex]\(0, 3, 5, 8\)[/tex]
We can see that the only element common to both [tex]\(D\)[/tex] and [tex]\(G\)[/tex] is [tex]\(3\)[/tex].
Thus, the intersection of [tex]\(D\)[/tex] and [tex]\(G\)[/tex] is:
[tex]\[ D \cap G = \{3\} \][/tex]
### 2. Find the Union of [tex]\(D\)[/tex] and [tex]\(G\)[/tex]
The union of two sets is the set of all elements that are in either set or in both sets, without any duplicates. To find the union, we combine all unique elements from both [tex]\(D\)[/tex] and [tex]\(G\)[/tex].
Combining the elements from both sets:
- From [tex]\(D\)[/tex]: [tex]\(-2, -1, 2, 3\)[/tex]
- From [tex]\(G\)[/tex]: [tex]\(0, 3, 5, 8\)[/tex]
List out all the unique elements from both sets:
[tex]\[ \{-2, -1, 2, 3, 0, 5, 8\} \][/tex]
When arranged in ascending order:
[tex]\[ \{-2, -1, 0, 2, 3, 5, 8\} \][/tex]
Thus, the union of [tex]\(D\)[/tex] and [tex]\(G\)[/tex] is:
[tex]\[ D \cup G = \{0, 2, 3, 5, 8, -1, -2\} \][/tex]
### Summary of Results
- The intersection of [tex]\(D\)[/tex] and [tex]\(G\)[/tex] is: [tex]\[\{3\}\][/tex]
- The union of [tex]\(D\)[/tex] and [tex]\(G\)[/tex] is: [tex]\[\{0, 2, 3, 5, 8, -1, -2\}\][/tex]
These are our answers in set notation.
