Question

Write the following sets by listing their elements between braces. $$ \mathscr{P}(\\{\mathbb{R}, \mathbb{Q}\\}) $$

Short answer

The elements of the power set \(\mathscr{P}(\{\mathbb{R}, \mathbb{Q}\})\) are \(\emptyset\), \(\{\mathbb{R}\}\), \(\{\mathbb{Q}\}\) and \(\{\mathbb{R}, \mathbb{Q}\}\).

Step-by-step solution

Recall the definition of the power set

The power set of a set S is the set of all possible subsets of S, including the empty set and the set itself. The notation used for the power set of S is usually \(\mathscr{P}(S)\). For example, if S is the set \(\{1, 2\}\), then the power set of S would be \(\{\emptyset, \{1\}, \{2\}, \{1, 2\}\}\). Taking the example given, the set S would be \(\{\mathbb{R}, \mathbb{Q}\}\) where \(\mathbb{R}\) is the set of all real numbers, and \(\mathbb{Q}\) is the set of all rational numbers.

Construct the Power Set

Now that the initial set S is identified, the power set can be constructed. We have to remember that for a set S with n elements, the power set will contain \(2^n\) elements or subsets. Since the set S consists of 2 elements (\(\mathbb{R}\) and \(\mathbb{Q}\)), the power set \(\mathscr{P}(S)\) consists of \(2^2 = 4\) subsets. The four subsets of S that make up the power set \(\mathscr{P}(S)\) are the empty set \(\emptyset\), \(\{\mathbb{R}\}\), \(\{\mathbb{Q}\}\) and \(\{\mathbb{R}, \mathbb{Q}\}\).

Key concepts

Set Theory

In mathematics, set theory is a fundamental theory that deals with collections of objects, known as sets. A set can be anything: a collection of numbers, letters, shapes, or even other sets. The beauty of set theory lies in its simplicity and universality; it provides a basic language through which a wide range of mathematical concepts can be expressed and explored.

The basic operations in set theory include union, intersection, and complement, while advanced topics delve into infinite sets, cardinality, and ordinal numbers. A set is defined by its elements, and each element is unique within that set. For instance, the set of all colors of a rainbow, the set of all prime numbers less than 10, or the set of all planets in our solar system. The definition of the set is crucial, as it determines which elements are included and which are not.

Subsets

A subset is a set that is entirely contained within another set. This is a core concept in set theory that emphasizes the 'part of' relationship between sets. For example, if we take a set A containing numbers {1, 2, 3, 4}, then {2, 3} would be a subset of A, as all elements of {2, 3} are also in A.

It is interesting to note that according to set theory, every set is considered a subset of itself and the empty set, denoted by \(\emptyset\), is a subset of every set, containing no elements. Subsets can be proper or improper. If a subset contains all the elements of the original set, it is called an improper subset, otherwise, it is called a proper subset. Understanding the concept of subsets is vital in exploring the power set, which consists of all possible subsets of the original set, including the set itself and the empty set.

Mathematical Notation

In the realm of mathematics, mathematical notation serves as a concise and efficient language. It is a system of symbols and signs used to express mathematical thoughts and ideas clearly and succinctly. The notation allows for the universal understanding of mathematical concepts, transcending language barriers.

For instance, the symbol \(\mathbb{R}\) represents the set of all real numbers, encompassing both rational and irrational numbers. On the other hand, \(\mathbb{Q}\) denotes the set of all rational numbers, which can be expressed as the quotient of two integers. Symbols such as \(\cup\) and \(\cap\) represent the union and intersection of sets, respectively. In the context of power sets, the notation \(\mathscr{P}(S)\) is commonly used to denote the power set of a set S, which is a set of all possible subsets of S. Mathematical notation is the foundation that allows students and professionals alike to communicate complex ideas in a digestible form.