If A = {a, (b,c)}. Find P(A).
Understand the Problem
The question is asking to find the power set P(A) of the set A, which contains the elements {a, (b, c)}. This involves determining all the subsets of the given set.
Answer
The power set \( P(A) \) is \( \{\emptyset, \{a\}, \{(b,c)\}, \{a, (b,c)\}\} \).
Answer for screen readers
The power set ( P(A) ) is: $$ P(A) = {\emptyset, {a}, {(b,c)}, {a, (b,c)}} $$
Steps to Solve
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Determine the number of elements in set A
Set ( A = {a, (b,c)} ) contains 2 elements: ( a ) and the subset ( (b,c) ). -
Calculate the number of subsets
The formula for the number of subsets of a set with ( n ) elements is ( 2^n ). Here, ( n = 2 ): $$ 2^2 = 4 $$ -
List all subsets of set A
The subsets of ( A ) include:
- The empty set: ( \emptyset )
- Each individual element: ( {a} ) and ( {(b,c)} )
- The entire set: ( {a, (b,c)} )
So the subsets are:
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( \emptyset )
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( {a} )
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( {(b,c)} )
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( {a, (b,c)} )
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Form the power set
The power set ( P(A) ) contains all the subsets we just found: $$ P(A) = {\emptyset, {a}, {(b,c)}, {a, (b,c)}} $$
The power set ( P(A) ) is: $$ P(A) = {\emptyset, {a}, {(b,c)}, {a, (b,c)}} $$
More Information
A power set is a set of all possible subsets of a given set, including the empty set and the set itself. The number of subsets grows exponentially with the number of elements in the original set.
Tips
- Miscounting the number of elements in the original set can lead to an incorrect number of subsets.
- Forgetting to include the empty set as a valid subset.
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