Problem 12 - Discrete Mathematics with Applications

Question

Determine if the given sets are equal. $$\\{x,\\{y\\}\\},\\{\\{x\\}, y\\}$$

Step-by-Step Solution

Answer

The given sets \(\{x, \{y\}\}\) and \(\{\{x\}, y\}\) are not equal because their elements do not match: \(x\) is not equal to \(\{x\}\) and \(\{y\}\) is not equal to \(y\).

Step 1: Observe the elements of the sets

For two sets to be equal, they must have the same elements. To check this, let's look at the elements in both sets: Set 1: \(\{x, \{y\}\}\) - Element 1: \(x\) - Element 2: \(\{y\}\) Set 2: \(\{\{x\}, y\}\) - Element 1: \(\{x\}\) - Element 2: \(y\)

Step 2: Compare the elements

Now that we have listed the elements separately for both sets, we can compare them: Set 1 element 1 (\(x\)) is not equal to Set 2 element 1 (\(\{x\}\)) because one is a set containing x and the other is the element x itself. Set 1 element 2 (\(\{y\}\)) is not equal to Set 2 element 2 (\(y\)) because one is a set containing y and the other is the element y itself.

Step 3: Determine if the sets are equal

Since the elements of Set 1 are not equal to the corresponding elements of Set 2, we can conclude that the sets are not equal.

Key Concepts

Set Notation

Exploring the language of sets is crucial for communicating mathematical ideas clearly. Set notation is a system used to describe collections of objects, which are referred to as 'elements'. When we talk about sets, we typically use curly braces, \( \{ \} \), to list the elements. For example, a set containing the elements a and b would be notated as \( \{a, b\} \). Each object in the set is unique; duplicates are ignored. Moreover, the order in which elements are listed is irrelevant.

There are some special notations as well. For instance, the empty set, a set with no elements, is denoted as \( \{othing\} \), or sometimes with the symbol \( \emptyset \). Understanding and using the correct set notation is fundamental when discussing concepts like set equality, as it sets the stage for accurately comparing and analyzing collections of elements.

Elements of a Set

Diving into sets and their constituents, we recognize that elements are the individual objects that make up a set. An element can be anything: a number, a letter, a symbol or even another set, which is known as a subset. In set theory, we emphasize that an element is a singular entity, whereas a set, even if it contains only one element, is still a collection.

In the language of sets, we denote that an element \(a\) is a member of set \(A\) by using the symbol \( \in \), which reads 'a is in A'. Conversely, if \(a\) is not in \(A\), we write \(a otin A\). For instance, in our given exercise, \(x\) is an element of the set \( \{x, \{y\}\} \), and \( \{y\} \) is not just an element of the first set but also a set by itself containing the element \(y\). Understanding the distinction between elements and sets within sets is key when analyzing the makeup of complex set structures.

Comparing Sets

Analyzing relationships between sets involves comparing elements and structures. When we say that two sets are equal, we imply that they contain the same elements, no more, no less, and without regard to order. Here, subtleties matter a great deal. For example, the sets \( \{1, 2\} \) and \( \{2, 1\} \) are considered equal because the same elements are present, despite being listed in a different order.

In our earlier exercise, however, the given sets appear similar at a glance, but they differ fundamentally. The set \( \{x, \{y\}\} \) includes an element \(x\) and a set \( \{y\}\) as its elements. In contrast, the set \( \{\{x\}, y\} \) includes a set \( \{x\}\) and an element \(y\). This subtle distinction—in how elements and subsets are contained within the larger set—meant that despite having 'similar' elements, they are not identical and therefore not equal. By understanding these minute details, one can cut through the apparent complexity and accurately perform set comparisons.