Question

If \(\mathrm{U}=\\{1,2,3\\}\) and \(\mathrm{A}=\\{1,2\\}\) then \([\mathrm{P}(\mathrm{A})]^{\prime}=\ldots \ldots \ldots\) (a) \(\\{\\{3\\},\\{2,3\\},\\{1,3\\},\\{1,2\\}, \Phi\\}\) (b) \(\\{\\{3\\},\\{2,3\\},\\{1,3\\},\\{1,2,3\\}\\}\) (c) \(\\{\\{3\\},\\{2,3\\},\\{1,3\\},\\{1,2,3\\}, \Phi\\}\) (d) \(\\{\\{3\\},\\{2,3\\},\\{1,3\\},\\{1,2\\}\\}\)

Short answer

(b) \(\{\{3\},\{2,3\},\{1,3\},\{1,2,3\}\}\)

Step-by-step solution

Finding the Power Set of A

First, we need to find the power set of A. The power set of A (\(\mathrm{P}(\mathrm{A})\)) includes all possible subsets of A, in this case: \(\Phi\), \(\{1\}\), \(\{2\}\), and \(\{1,2\}\). So, \(\mathrm{P}(\mathrm{A})\) = \(\{\Phi, \{1\}, \{2\}, \{1,2\}\}\).

Finding the Power Set of U

Next, we need to find the power set of the universe U (denoted as \(\mathrm{P}(\mathrm{U})\)). U = \(\{1, 2, 3\}\), so the possible subsets are: \(\Phi\), \(\{1\}\), \(\{2\}\), \(\{3\}\), \(\{1,2\}\), \(\{1,3\}\), \(\{2,3\}\), and \(\{1,2,3\}\). Thus, \(\mathrm{P}(\mathrm{U})\) = \(\{\Phi, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}\).

Finding the Complement of Power Set of A

To find the complement of the power set of A, denoted as \([\mathrm{P}(\mathrm{A})]^{\prime}\), we will take the elements of \(\mathrm{P}(\mathrm{U})\) which are not in \(\mathrm{P}(\mathrm{A})\). So, \([\mathrm{P}(\mathrm{A})]^{\prime}\) = \(\{\{3\}, \{1,3\}, \{2,3\},\{1,2,3\}\}\). Comparing our result with the given options, we find the correct answer is (b) \(\{\{3\},\{2,3\},\{1,3\},\{1,2,3\}\}\).

Key concepts

Power Set

In set theory, the concept of the power set is fundamental. The power set of any given set is the set of all possible subsets of that set, including the empty set and the set itself. Suppose we have a set \(A = \{1, 2\}\). The power set of \(A\), denoted as \(\mathrm{P}(A)\), includes:

• \(\{ \} \) - the empty set
• \(\{1\}\)
• \(\{2\}\)
• \(\{1, 2\}\) - the set itself

Thus, \(\mathrm{P}(A) = \{\{ \}, \{1\}, \{2\}, \{1, 2\}\}\). This collection represents all the different ways the elements of \(A\) can be grouped.
Creating a power set is an exercise in considering every combination of set elements.

Universal Set

The universal set is a set containing all the objects under consideration for a particular discussion or problem. It is often denoted by \(U\). For example, if \(U = \{1, 2, 3\}\), it means that these are all the elements we consider in a given context.
The importance of the universal set becomes evident when we are finding complements of sets or doing set operations like union and intersection.

• The universal set includes every possible element, allowing us to define the idea of set complements more effectively.

In the problem at hand, \(U\) was used to explore the elements that were not included in the power set of \(A\).

Set Complement

A set complement refers to the set of all elements in the universal set that are not in the given set. If \(A\) is a subset of \(U\), the complement of \(A\), denoted \(A'\), includes all elements from \(U\) not found in \(A\).
To locate the complement of the power set of \(A\), we find subsets in the power set of \(U\) that are missing in the power set of \(A\).

• If \(\mathrm{P}(A) = \{\{ \}, \{1\}, \{2\}, \{1, 2\}\}\)
• And \(\mathrm{P}(U) = \{\{ \}, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\}\)

Then, the complement \([\mathrm{P}(A)]^{\prime}\) is \(\{\{3\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\}\). This set complements \(\mathrm{P}(A)\) within \(\mathrm{P}(U)\).

Subsets

A subset is a set whose elements are all contained within another set. If \(A\) is a subset of \(B\), then every element of \(A\) is also an element of \(B\).

• This relationship is crucial when forming power sets, as each element of a power set is itself a subset of the original set.

Consider set \(A = \{1, 2\}\). The subsets of \(A\) include:

• The empty set \(\{ \}\)
• The set containing just \(1\)
• The set containing just \(2\)
• The set \(A\) itself \(\{1, 2\}\)

Understanding subsets helps in grasping how power sets build combinations, and how each combination fits into larger discussions about set complement and universality within set theory.