Understanding the Given Set A

The question asks for the power set of the given set A. The set A is defined as \(A = \{\lambda, \{\lambda, \mu\}\}\).

To find the power set of A, we first need to identify the elements of A. Looking at the definition of A, we can see that A contains two elements:

• The first element is \(\lambda\).
• The second element is the set \(\{\lambda, \mu\}\).

Let's denote the elements of A as \(e_1 = \lambda\) and \(e_2 = \{\lambda, \mu\}\). So, \(A = \{e_1, e_2\}\).

The number of elements in set A is \(|A| = 2\).

Calculating the Power Set of A

The power set of a set is the set of all possible subsets of that set. If a set has \(n\) elements, its power set will have \(2^n\) elements.

Since set A has \(|A| = 2\) elements, its power set, denoted as \(P(A)\), will have \(2^2 = 4\) elements.

The elements of the power set are the subsets of A. Let's list all possible subsets of \(A = \{\lambda, \{\lambda, \mu\}\}\):

1. The empty set: The empty set (\(\phi\) or \{\}) is a subset of every set.
2. Subsets containing one element:
   • The set containing the first element of A: \(\{\lambda\}\)
   • The set containing the second element of A: \(\{\{\lambda, \mu\}\}\)
3. Subsets containing two elements:
   • The set containing both elements of A: \(\{\lambda, \{\lambda, \mu\}\}\) (This is the set A itself)

Combining all these subsets, the power set of A is:

\(P(A) = \{\phi, \{\lambda\}, \{\{\lambda, \mu\}\}, \{\lambda, \{\lambda, \mu\}\}\}\)

Comparing with the Given Options

Now let's compare our calculated power set with the provided options:

• Option 1: \(\{\phi, \{\phi\}, \{\lambda\}, \{\lambda, \mu\}\}\). This option includes \(\{\phi\}\) which is not a subset of A, and \(\{\lambda, \mu\}\) which is the second element itself, not a subset containing only the second element. Incorrect.
• Option 2: \(\{\phi, \{\lambda\}, \{\{\lambda, \mu\}\}, \{\lambda, \{\lambda, \mu\}\}\}\). This matches our calculated power set exactly. It includes the empty set, the subset containing only the first element, the subset containing only the second element (which is the set \(\{\lambda, \mu\}\)), and the subset containing both elements (which is A). Correct.
• Option 3: \(\{\phi, \{\lambda\}, \{\lambda, \mu\}, \{\lambda, \{\lambda, \mu\}\}\}\). This option includes \(\{\lambda, \mu\}\) as a subset. However, \(\{\lambda, \mu\}\) is an element of A, not a subset formed from elements of A (except as the single element subset \(\{\{\lambda, \mu\}\}\) or as part of the full set A). Incorrect.
• Option 4: \(\{\{\lambda\}, \{\lambda, \mu\}, \{\lambda, \{\lambda, \mu\}\}\}\). This option is missing the empty set \(\phi\) and includes \(\{\lambda, \mu\}\) incorrectly as a subset. Incorrect.

Based on the comparison, Option 2 correctly represents the power set of \(A = \{\lambda, \{\lambda, \mu\}\}\).

Revision Table: Key Concepts

Additional Information on Power Sets and Set Elements

It is important to distinguish between an element of a set and a subset containing that element. For the set \(A = \{\lambda, \{\lambda, \mu\}\}\):

• \(\lambda\) is an element of A (\(\lambda \in A\)). \(\{\lambda\}\) is a subset of A (\(\{\lambda\} \subseteq A\)).
• \(\{\lambda, \mu\}\) is an element of A (\(\{\lambda, \mu\} \in A\)). \(\{\{\lambda, \mu\}\}\) is a subset of A (\(\{\{\lambda, \mu\}\} \subseteq A\)). This subset contains the single element \(\{\lambda, \mu\}\).

The empty set \(\phi\) (or \{\}) is always a subset of any set, including A (\(\phi \subseteq A\)). The set A itself is also always a subset of A (\(A \subseteq A\)).

When constructing the power set, we collect all these subsets as the elements of the power set. The power set is a set of sets.