Questions: A=1,8,5,9,2,3,0,6 A= Ex: 5 P(A)= Ex: 5

Transcript text: A=\{1,8,5,9,2,3,0,6\} \\ |A|=\text { Ex: } 5: \\ |P(A)|=\text { Ex: } 5

Solution

To solve the problem, we need to understand the following concepts:

Set Cardinality: The cardinality of a set \( A \), denoted \( |A| \), is the number of elements in the set.
Power 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 is \( 2^{|A|} \).

Given the set \( A = \{1, 8, 5, 9, 2, 3, 0, 6\} \), we will calculate the cardinality of \( A \) and the cardinality of its power set.

Step 1: Determine the Set and Its Cardinality

Given the set \( A = \{0, 1, 2, 3, 5, 6, 8, 9\} \), we find the cardinality of \( A \) as follows: \[ |A| = 8 \]

Step 2: Calculate the Cardinality of the Power Set

The cardinality of the power set \( P(A) \) is calculated using the formula: \[ |P(A)| = 2^{|A|} = 2^8 = 256 \]

The cardinality of the set \( A \) is \( |A| = 8 \) and the cardinality of the power set \( P(A) \) is \( |P(A)| = 256 \).

Thus, the final answers are: \[ \boxed{|A| = 8} \] \[ \boxed{|P(A)| = 256} \]

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