Final answer:
The power set of a set is the set of all its subsets, including the empty set and the set itself. In this case, the given set contains two elements: the empty set (∅) and a set containing only the empty set, which can be written as {∅}. To find the power set, we need to list all the possible combinations of these two elements, including the empty set and the set itself, thus the correct option is C.
Step-by-step explanation:
The power set of a set is the set of all its subsets, including the empty set and the set itself. In this case, the given set contains two elements: the empty set (∅) and a set containing only the empty set, which can be written as {∅}. To find the power set, we need to list all the possible combinations of these two elements, including the empty set and the set itself.
First, we start by listing the empty set and the set itself, as these are always part of the power set. This gives us {∅, {∅}}. Next, we need to consider all the possible combinations of these two elements, which are: {∅, {∅}}, {{∅}}, and {{∅}, {∅}}. However, we can simplify this by noticing that the set {{∅}} is the same as the set {∅}. This is because both sets contain only the empty set, so they are essentially the same set. Therefore, we can eliminate {{∅}} from our list and our final answer becomes {∅, {∅}, {{∅}}, {∅, {{∅}}}.
Step-by-step explanation:
To understand why the power set of {∅, {∅}} is {∅, {∅}, {{∅}}, {∅, {{∅}}}}, we need to understand the concept of subsets and how they are related to the power set. A subset is a set that contains elements from a larger set. In this case, the elements of our larger set are the empty set (∅) and a set containing only the empty set, which is {∅}. Therefore, the subsets of this larger set can be formed by selecting either one or both of these elements. This is why our final answer contains all the possible combinations of these two elements, including the empty set and the set itself.
Let's break down each element in our final answer to understand their significance. The first element, ∅, represents the empty set. This is always part of the power set as every set contains the empty set as a subset. The second element, {∅}, represents the set containing only the empty set. This is also always part of the power set as every set is a subset of itself. The third element, {{∅}}, represents a set containing only one subset, which is the empty set. This is also always part of the power set. The last element, {∅, {{∅}}}, represents a set containing two subsets: the empty set and the set containing only the empty set. This is the only element that contains more than one subset and is therefore, also part of the power set.
In summary, the power set of {∅, {∅}} is {∅, {∅}, {{∅}}, {∅, {{∅}}}} because it includes all the possible combinations of the elements of the given set, including the empty set and the set itself. It is important to note that the order of the elements in the power set does not matter, which is why we can write {∅, {∅}, {{∅}}, {∅, {{∅}}}} in any order and it will still be considered the correct answer, thus the correct option is C.