91Ó°ÊÓ

Set theory is a fundamental area of mathematics that deals with collections of objects, called sets. In this exercise, we work with a set of lists formed from distinct integers pulled from the set \(1, 2, \backslash ldots, n\). An important concept in set theory is how sets relate to each other.

Here, two lists are considered related if they share the same elements under certain conditions. These conditions are met when they have identical elements in the first \(k\) positions and identical elements in the last \(n-k\) positions, though the order within those segments may differ.

Understanding how these lists are classified into related groups is key to grasping the concept of equivalence relations, a crucial notion within set theory.

Reflexivity is a property of a relation where every element is related to itself. To show reflexivity for our relation \(R\), we need to prove that any given list \L\ is related to itself.

Consider any list \L\. By definition, \L\ has the same elements in its first \(k\) places as itself. Similarly, it has the same elements in its last \(n-k\) places as itself. Therefore, \L \ R \ L\, proving that the relation is reflexive for every list in our set.

Reflexivity tells us self-comparison, a crucial check in validating that a relation qualifies as an equivalence relation.

Symmetry is another essential property of equivalence relations where if one element is related to a second, the second is related to the first. For our relation \(R\), we need to verify that if two lists \L_{1} \ and \L_{2} \ are related, then \L_{2} \ is also related to \L_{1} \.

Assume \L_{1} \ R \ L_{2} \. This means \L_{1} \ and \ L_{2} \ share identical elements in both their first \(k\) places and their last \(n-k\) places. Since the order of elements in a set does not affect their equality, \L_{2} \ must also have the same elements as \L_{1} \ in these positions. Therefore, \ L_{2} \ R \ L_{1} \.

The symmetry property is successfully shown, adding one more step in characterizing \(R\) as an equivalence relation.

Transitivity is the last essential property we need to establish for an equivalence relation. A relation is transitive if, whenever \L_{1} \ is related to \L_{2} \, and \L_{2} \ is related to \L_{3} \, then \ L_{1} \ is also related to \L_{3} \.

Suppose \L_{1} \ R \ L_{2} \ and \L_{2} \ R \ L_{3} \. This means \L_{1} \ and \L_{2} \ share the same elements in their first \(k\) places and last \(n-k\) places. Similarly, \L_{2} \ and \L_{3} \ share the same elements in those segments. Consequently, \ L_{1} \ and \L_{3} \ must have identical elements in their first \(k\) places and last \(n-k\) places. Thus, \L_{1} \ R \ L_{3} \.

By establishing transitivity, we complete our verification that the relation \(R\) is indeed an equivalence relation.