To determine if a set [tex]\( D = \{-2, -1, 0, 1, 2\} \)[/tex] is a subset of itself, we need to understand the definition of a subset.
A set [tex]\( A \)[/tex] is considered a subset of set [tex]\( B \)[/tex] if every element of [tex]\( A \)[/tex] is also an element of [tex]\( B \)[/tex]. In other words, for all [tex]\( x \in A \)[/tex], it must be true that [tex]\( x \in B \)[/tex].
In the case where we are examining whether [tex]\( D \)[/tex] is a subset of itself, we have:
- [tex]\( A = D \)[/tex]
- [tex]\( B = D \)[/tex]
By the definition of a subset, for [tex]\( D \)[/tex] to be a subset of [tex]\( D \)[/tex], all elements in [tex]\( D \)[/tex] must be contained within [tex]\( D \)[/tex].
Looking at the set [tex]\( D \)[/tex]:
- The elements of [tex]\( D \)[/tex] are \{-2, -1, 0, 1, 2\}.
Since every element in [tex]\( D \)[/tex] is naturally present in [tex]\( D \)[/tex], it follows that every element of [tex]\( D \)[/tex] is also an element of [tex]\( D \)[/tex].
Thus, [tex]\( D \)[/tex] is a subset of itself.
Therefore, the answer is:
[tex]\[ \text{Yes, } D \text{ is a subset of itself.} \][/tex]
This conclusion aligns with the properties of sets in mathematics, where any set is always considered a subset of itself.
