,Final answer:
The correct answer is options (a) and (c), which are [x∈ℝ:x≤2],[x∈ℝ:2≤x<4),[x∈ℝ:4≤x) and [x∈ℝ:x<2),[x∈ℝ:2≤x<4),[x∈ℝ:4≤x] respectively, as they both represent collections of sets that form a partition of the real numbers, with no intersections and covering all possible values.
Step-by-step explanation:
The question asks to select the collection of sets that forms a partition of the real numbers, ℝ. A partition of a set is a collection of disjoint subsets that together cover the original set without any overlaps. In other words, every element in the original set is included in exactly one of the subsets. Considering the options provided:
- Option a: [x∈ℝ:x≤2],[x∈ℝ:2≤x<4),[x∈ℝ:4≤x) forms a partition because each real number is included in exactly one subset: numbers less than or equal to 2, numbers between 2 (inclusive) and 4 (exclusive), and numbers greater than or equal to 4.
- Option b: [x∈ℝ:x<4),(x∈ℝ:25x≤4),[x∈ℝ:2 does not form a partition since it includes typos and is unclear.
- Option c: [x∈ℝ:x<2),[x∈ℝ:2≤x<4),[x∈ℝ:4≤x] also forms a partition because it includes all real numbers, with sets that cover numbers less than 2, between 2 and 4, and greater than or equal to 4 without overlapping.
- Option d: [x∈ℝ:x<2),[x∈ℝ:2,[x∈ℝ:4≤x) is not forming a partition due to the missing conditions and unclear intervals.
Therefore, the correct collections of sets that forms a partition of ℝ are the options (a) and (c).