Question

Let \(S=\\{1,2,3,4\\}\). The total number of unordered pairs of disjoint subsets of \(S\) is equal to A) 25 B) 34 C) 42 D) 41

Short answer

23 unordered pairs of disjoint subsets

Step-by-step solution

Understanding Disjoint Subsets

Two subsets, A and B, of a set S are called 'disjoint' if they have no elements in common. This means that for every element in A, it is not also in B, and vice versa.

Enumerating Disjoint Subsets of a Single Element

Looking at the single element scenarios, where one subset contains a single element and the other contains none or the other elements, we realize that for each element in S, there cannot be an overlap. For each of the 4 elements in S, we get 3 disjoint subsets (the individual element and one of each of the remaining elements), totaling 12 disjoint single-element pairs.

Consider Empty Set Scenarios

We must also consider the empty set as a subset. Each element in S can be paired with an empty set. This gives us 4 more disjoint pairs, one for each element.

Consider Multiple Elements and Disjointness

We have to consider subsets with multiple elements. There are 6 ways to make pairs of subsets where both subsets have two unique elements (for example, {1,2} and {3,4}). There are no possible disjoint subsets where one has three elements because the other would necessarily share at least one element.

Counting the Total Disjoint Subsets

Add up the disjoint subsets from step 2, step 3, and step 4. There are 12 single-element disjoint pairs, 4 involving the empty set and 6 pairs of two-element subsets. This sums up to 12 + 4 + 6 = 22 pairs.

Include the Pair of Two Empty Sets

Don't forget to include the pair of two empty sets as disjoint subsets because the empty set is also a subset of S.

Computing the Final Answer

Finally, add the pair of empty sets to the 22 previously counted pairs to arrive at a total of 22 + 1 = 23 unordered pairs of disjoint subsets.

Key concepts

Combinatorics and the Disjoint Subsets Problem

Combinatorics is a branch of mathematics that deals with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications in computer science, statistics, and physics. The disjoint subsets problem, which is a common question type in combinatorics, requires us to understand how to count combinations of objects (in this case, sets) that meet a certain criterion (here, being 'disjoint' or not sharing any elements).

In this exercise, we look at the subsets of the set \( S=\{1,2,3,4\} \). We're asked to determine how many unordered pairs of disjoint subsets we can form from \( S \). The solution to this puzzle exemplifies the application of basic combinatorics principles. Each element of the set \( S \) can form a subset that is disjoint with several other subsets. Remembering that disjoint subsets have no elements in common, we can carefully enumerate the possible combinations or pairs without repetition.

Combinatorics problems often require the method of counting without overcounting, which is a key in solving such exercises. It is essential to consider different scenarios and count the possibilities methodically as demonstrated in the step-by-step solution. Such problems are integral in competitive exams like JEE Advanced mathematics where combinatorics plays a significant role.

Set Theory Fundamentals

Set theory is the mathematical theory of well-determined collections, called sets, of objects that are called members, or elements, of the set. In set theory, a subset is a set that is fully contained within another set, and an empty set is a set with no elements, denoted as \( \emptyset \).

The concept of disjoint subsets—subsets with no element in common—is fundamental in set theory and is exemplified in this problem. When solving for disjoint subsets, it's helpful to visualize sets and their subsets, noting which elements are included and which are not. For instance, the subsets \( \{1,2\} \) and \( \{3,4\} \) are disjoint because they share no common elements. A crucial aspect to consider is the empty set, which is disjoint with any other subset since it doesn't contain any elements at all.

The empty set often leads to overlooked combinations, especially in problems that require careful enumeration like the one in the textbook exercise. Remember that set theory not only deals with the existence of sets and subsets but also with operations on sets, which include intersection (common elements), union (all elements), and complement (elements not in the subset) that are also often tested in competitive exams like JEE Advanced.

JEE Advanced Mathematics and the Importance of Disjoint Subsets

The Joint Entrance Examination (JEE) Advanced is known for testing the depth of understanding and the application of concepts rather than just rote learning. It covers a broad range of mathematics topics, including algebra, trigonometry, calculus, and especially, combinatorics and set theory. The problem given is a type that may appear in the JEE Advanced, specifically under the combinatorics and set theory section.

In this context, students are challenged to apply their theoretical knowledge to practical problems. Identifying pairs of disjoint sets requires a strong foundational understanding and an ability to apply the principles of combinatorics and set theory to count appropriate combinations correctly. The examination emphasises problem-solving abilities which are apparent from the incremental steps provided in the solution. Each step builds upon the previous, from considering single elements to evaluating the significance of an empty set.

Enhancing the problem-solving skills necessary for the disjoint subsets problem involves practicing various combinatorics exercises and understanding set theory deeply. For students aspiring to crack the JEE Advanced, becoming proficient in these topics is not only beneficial for the exam but also for future studies in fields that value mathematical reasoning and discrete mathematics.