Question

List all the subsets of the following sets. $$ \\{\mathbb{R}, \mathbb{Q}, \mathbb{N}\\} $$

Short answer

The all possible subsets of the set {ℝ, ℚ, ℕ} are {}, {ℝ}, {ℚ}, {ℕ}, {ℝ, ℚ}, {ℝ, ℕ}, {ℚ, ℕ}, and {ℝ, ℚ, ℕ}.

Step-by-step solution

Understand What a Subset Is

A subset is a part of a set, including the possibility of the set itself and also an empty set. Any group of elements that are a part of the original set, regardless of their order or their count (may or may not be together), is considered a subset of the original set.

List All Subsets

The full list of subsets of the given set {ℝ, ℚ, ℕ} is:\n\n- The empty set {}\n- {ℝ}\n- {ℚ} \n- {ℕ}\n- {ℝ, ℚ} \n- {ℝ, ℕ}\n- {ℚ, ℕ}\n- The set itself {ℝ, ℚ, ℕ}

Key concepts

Set Theory

Set theory is the mathematical study of collections of objects, which are referred to as 'sets'. It's a foundational system for mathematics, underpinning various aspects of the discipline. In set theory, a set is defined by its elements, which can be anything: numbers, symbols, points in space, or even other sets. The notion is fairly straightforward - if you have a collection of distinct objects, you have a set.

For example, the set of natural numbers \( \mathbb{N} \) is a collection that includes 1, 2, 3, and so on. But set theory isn't just about simple collections; it also involves the relations between sets, operations on sets such as union, intersection, and difference, and conceptualizing mathematical objects in terms of set-theoretical concepts.

A key idea in set theory is membership; if an object belongs to a set, it is considered a 'member' or 'element' of that set. Each element can only be counted once within a set, emphasizing the importance of unique elements.

Mathematical Sets

In mathematics, sets are the fundamental building blocks for various types of numbers and more abstract entities. A set can be defined explicitly by listing its elements or described implicitly by a shared property of its elements. For instance, the set of all even numbers is implicitly defined by the property that all elements are divisible by two.

Sets are often denoted using curly braces, as in \( \{1, 2, 3\} \). A vital concept to understand is that sets are unordered; the set \( \{3, 2, 1\} \) is the same as \( \{1, 2, 3\} \). In the original exercise, we are presented with a set containing the sets of real numbers \( \mathbb{R} \), rational numbers \( \mathbb{Q} \), and natural numbers \( \mathbb{N} \). It's essential to recognize that these mathematical sets can contain an infinite number of elements and can even include other sets as elements.

Power Set

A power set is a concept from set theory that represents all possible subsets of a set, including the set itself and the empty set. For any given set \( S \), its power set is denoted as \( \mathcal{P}(S) \) and includes every combination of its elements. \

\ For example, if our set \( S \) is \( \{a, b\} \), the power set of \( S \) would be \( \{\emptyset, \{a\}, \{b\}, \{a, b\}\} \). The power set of a set with \( n \) elements will contain \( 2^n \) elements, as each element can either be included or excluded from each subset. This exponential relationship shows how quickly the complexity of a power set increases with the size of the original set. To understand the exercise solution better, students should grasp that the power set includes all possible subsets, which is what the step-by-step solution lists for the given set of \( \mathbb{R}, \mathbb{Q}, \mathbb{N} \).