Let's address the given problems one by one:
### (a) List the elements of the set B.
The set [tex]\( B \)[/tex] consists of natural numbers less than 5. Natural numbers start from 1, so the numbers less than 5 are 1, 2, 3, and 4. Therefore, the elements of set [tex]\( B \)[/tex] are:
[tex]\[ B = \{1, 2, 3, 4\} \][/tex]
### (b) Write whether A and B are equal or equivalent sets.
Two sets are considered equal if they contain exactly the same elements. In this case:
[tex]\[ A = \{2, 4, 6, 8\} \][/tex]
[tex]\[ B = \{1, 2, 3, 4\} \][/tex]
Clearly, [tex]\( A \)[/tex] and [tex]\( B \)[/tex] contain different elements, so they are not equal sets.
Two sets are considered equivalent if they have the same number of elements. The sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] both contain 4 elements:
[tex]\[ |A| = 4 \][/tex]
[tex]\[ |B| = 4 \][/tex]
Since the number of elements in each set is the same, sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are equivalent. Therefore:
- [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not equal.
- [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are equivalent.
### (c) Are the sets A and B disjoint or overlapping? Give reason.
Two sets are disjoint if they have no elements in common. Conversely, if they share at least one common element, they are overlapping.
Let's examine if [tex]\( A \)[/tex] and [tex]\( B \)[/tex] share any common elements:
- Set [tex]\( A = \{2, 4, 6, 8\} \)[/tex]
- Set [tex]\( B = \{1, 2, 3, 4\} \)[/tex]
Both sets share the elements 2 and 4. Therefore, they are not disjoint because they have elements in common. Hence, the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are overlapping.
To summarize:
- Elements of set B: [tex]\[ \{1, 2, 3, 4\} \][/tex]
- Equality & Equivalence: A and B are not equal but equivalent.
- Disjoint or Overlapping: A and B are overlapping, as they share common elements.
### Additional Information
Sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are both finite since they contain a limited number of elements. This is evidenced by the fact that they have explicitly listed elements with a known count (4 elements each). A finite set has a countable number of elements, unlike an infinite set which extends indefinitely.
