Questions: A = 9,8,0,2,7 A = Ex: 50 P(A) = EX: 5

Transcript text: \[ \begin{aligned} A & =\{9,8,0,2,7\} \\ |A| & =\text { Ex: } 50 \\ |P(A)| & =\text { EX: } 5 \end{aligned} \]

Solution

Solution Steps

To solve the problem of finding the cardinality of a set \( A \) and its power set \( P(A) \), we need to understand the following:

The cardinality of a set \( A \), denoted as \(|A|\), is the number of elements in the set.
The power set \( P(A) \) of a set \( A \) is the set of all subsets of \( A \), including the empty set and \( A \) itself.
The cardinality of the power set \(|P(A)|\) is \(2^{|A|}\).

Given the set \( A = \{9, 8, 0, 2, 7\} \), we can calculate \(|A|\) and \(|P(A)|\).

Solution Approach

Determine the number of elements in set \( A \).
Calculate the power set's cardinality using the formula \(2^{|A|}\).

Step 1: Determine the Cardinality of Set \( A \)

The set \( A \) is defined as \( A = \{0, 2, 7, 8, 9\} \). The cardinality of set \( A \), denoted as \( |A| \), is the number of elements in the set. Thus, we have: \[ |A| = 5 \]

Step 2: Calculate the Cardinality of the Power Set \( P(A) \)

The power set \( P(A) \) is the set of all subsets of \( A \). The cardinality of the power set is given by the formula: \[ |P(A)| = 2^{|A|} \] Substituting the value of \( |A| \): \[ |P(A)| = 2^5 = 32 \]