Question

Find the following cardinalities. $$ |\\{\\{1,4\\}, a, b,\\{\\{3,4\\}\\},\\{\varnothing\\}\\}| $$

Short answer

The cardinality of the set is 5.

Step-by-step solution

Identify the elements

First, identify all the elements of the set. Each element is separated by a comma. In this set, there are five elements: \(\{1,4\}\), \(a\), \(b\), \(\{\{3,4\}\}\), and \(\{\varnothing\}\).

Count the elements

Now simply count the number of elements identified in step 1. There are a total of 5 elements in this set.

Key concepts

Set Theory

When we delve into the study of set theory, we are looking at the foundation of modern mathematics. Sets are collections of distinct objects, called elements, and they are fundamental to various mathematical disciplines. The concept is simple yet expansive: a set can contain numbers, letters, symbols, or even other sets.

In set theory, the focus isn't on what the elements are, but rather on the properties they hold and the relationships between them. Operations such as union, intersection, and complement help us understand and visualize the connections and distinctions between different sets. Importantly, cardinality is a measure of how many elements are in a set – this is essentially the set's size or length.

Understanding Cardinality

In the given exercise, determining the cardinality of a set involves counting the unique elements it contains. It does not matter if the elements themselves are numbers, symbols, or even sets – each counts as one towards the total cardinality.

Elements of a Set

The elements of a set are the individual items that make up the set. These elements can be anything: numbers, letters, other sets, or none (indicated by the symbol for the empty set, \(\varnothing\)). What's crucial is that each element is unique within the set; duplicates are not counted multiple times.

For example, in a set containing \(\{1, 2, 2, 3\}\), the actual elements are 1, 2, and 3. Even though '2' appears twice, it only counts once because sets are defined by unique values. When it comes to sets that contain other sets as elements (such as \(\{\{1,4\}\}\) from the exercise), these are called singleton sets and are treated as a single distinct element of the 'parent' set.

Singleton Sets and Nested Elements

In the context of set theory, recognizing these nested structures within a set allows for a clearer understanding of the concept of cardinality. Each singleton or nested set, no matter how complex, counts as one element in the determination of a set's cardinality.

Mathematical Logic

At its core, mathematical logic is about applying reasoning to mathematics. It's the branch that deals with formal systems and symbolic reasoning, often focusing on the consistency, completeness, and correctness of mathematical statements and proofs.

When solving set problems, such as finding cardinalities, mathematical logic is used to rigorously define the steps in the process. Each step in a solution has to logically follow from the last. An important aspect of mathematical logic applied to set theory is ensuring that the reasoning behind counting elements is both sound and clear.

For instance, each element within a set is treated individually when applying logic to determine the cardinality. Taking the set from the exercise, even though \(\{1,4\}\) and \(\{\{3,4\}\}\) might seem related because they both contain the number 4, logical rules stipulate that we count them separately due to their distinct presences as elements.