To solve the problem, let's carefully consider the intervals for the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] and perform the operations step by step.
Given:
[tex]\[ A = [-3, 2) \][/tex]
[tex]\[ B = (-2, 3) \][/tex]
### a) Finding [tex]\( A \cup B \)[/tex]
The union of two sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted as [tex]\( A \cup B \)[/tex], includes all elements that are in [tex]\( A \)[/tex], or in [tex]\( B \)[/tex], or in both.
- The interval [tex]\( A = [-3, 2) \)[/tex] contains all numbers from [tex]\(-3\)[/tex] to [tex]\(2\)[/tex], including [tex]\(-3\)[/tex] but not including [tex]\(2\)[/tex].
- The interval [tex]\( B = (-2, 3) \)[/tex] contains all numbers from [tex]\(-2\)[/tex] to [tex]\(3\)[/tex], not including [tex]\(-2\)[/tex] but including up to [tex]\(3\)[/tex].
To determine [tex]\( A \cup B \)[/tex], we need the combined range of both intervals without duplicating any elements, including the entire overlap:
1. The leftmost boundary of [tex]\( A \)[/tex] is [tex]\(-3\)[/tex], and the leftmost element of [tex]\( B \)[/tex] starts after [tex]\(-2\)[/tex]. Thus, the left boundary of the union is [tex]\(-3\)[/tex].
2. The rightmost boundary of [tex]\( B \)[/tex] is [tex]\(3\)[/tex], and the right boundary of [tex]\( A \)[/tex] stops at [tex]\(2\)[/tex], just before inclusively reaching it. Therefore, the right boundary of the union is [tex]\(3\)[/tex] because it includes [tex]\(3\)[/tex].
Thus, the union [tex]\( A \cup B \)[/tex] is the interval from [tex]\(-3\)[/tex] to [tex]\(3\)[/tex], including [tex]\(-3\)[/tex] but not including [tex]\(3\)[/tex]:
[tex]\[ A \cup B = [-3, 3) \][/tex]
### b) Finding [tex]\( A \cap B \)[/tex]
The intersection of two sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted as [tex]\( A \cap B \)[/tex], includes all elements that are common to both sets.
- The interval [tex]\( A = [-3, 2) \)[/tex] overlaps with [tex]\( B = (-2, 3) \)[/tex], requiring us to find where both intervals share common elements.
To determine [tex]\( A \cap B \)[/tex]:
1. The elements in [tex]\( A \)[/tex] that overlap with [tex]\( B \)[/tex] need to start from the right edge of [tex]\(-2\)[/tex] (since [tex]\(-3 < -2\)[/tex]).
2. Since [tex]\( A \)[/tex] continues until just before [tex]\(2\)[/tex] (exclusive), and [tex]\( B \)[/tex] continues beyond [tex]\(2\)[/tex], the rightmost shared boundary is [tex]\(2\)[/tex]. Hence, we stop at [tex]\(2\)[/tex] not including it.
Thus, the intersection [tex]\( A \cap B \)[/tex] is the interval:
[tex]\[ A \cap B = (-2, 2) \][/tex]
### Final Results:
a) The union [tex]\( A \cup B = [-3, 3) \)[/tex]
b) The intersection [tex]\( A \cap B = (-2, 2) \)[/tex]
