The set union of intervals [a, b] and [c, d], where a < b < c < d, can be expressed as the difference of two sets: [a, d] \setminus (b, c). This formulation highlights that the union includes all numbers from the interval [a, d] except those in the open interval (b, c). The discussion emphasizes that the intervals [a, b] and [c, d] are disjoint, confirming that (b, c) is a non-null set that can be subtracted to achieve the desired union.
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rayne said:Let a, b, c, and d be real numbers with a < b < c < d. Express the set [a, b]U[c, d] as the difference of two sets.
I know that [a,b]U[c,d] is a union and what a difference of two sets is, but I don't quite understand this question.
mathbalarka said:Note that you're given the condition $a < b < c < d$. Thus, $[a, b]$ and $[c, d]$ are disjoint, i.e., $(b, c)$ is non-null. Consider the compactfied interval $[a, d] \supset [a, b] \cup [c, d]$. Can you "subtract" something from $[a, d]$ to produce $[a, b] \cup [c, d]$?
And, of course, $[a, b]\cup[c, d]=([a, b]\cup[c, d])\setminus\emptyset$. Many other variants are possible. If the problem asked to represent $[a, b]\cup[c, d]$ as the difference of two segments, then I believe there is only one solution.rayne said:Let a, b, c, and d be real numbers with a < b < c < d. Express the set [a, b]U[c, d] as the difference of two sets.