The discussion revolves around expressing the union of two intervals, [a, b] and [c, d], as the difference between two sets. Participants explore the implications of the condition a < b < c < d and consider various ways to represent the union through set differences. The scope includes mathematical reasoning and conceptual clarification regarding set operations.
Participants generally agree on the method of expressing the union as a difference of sets, specifically using the intervals [a, d] and (b, c). However, there is some ambiguity regarding the uniqueness of the representation, as different participants suggest alternative formulations.
Some participants highlight the importance of the specific conditions given (a < b < c < d) in determining the validity of the proposed set differences. The discussion does not resolve whether there are other valid representations beyond those mentioned.
Readers interested in set theory, mathematical reasoning involving intervals, or those exploring the properties of unions and differences of sets may find this discussion beneficial.
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.