2. Determine the power set of 𝐴𝐴 = {{𝑎, 𝑏},{𝑐},{𝑑, 𝑒, 𝑓}}
The power set of a set 𝐴 is the set of all possible subsets of 𝐴, including the empty set and the set itself.
In this case, 𝐴 = {{𝑎, 𝑏},{𝑐},{𝑑, 𝑒, 𝑓}}.
To find the power set, we consider all possible combinations of the elements in 𝐴.
The elements of 𝐴 are:
Subset 1: {𝑎, 𝑏}
Subset 2: {𝑐}
Subset 3: {𝑑, 𝑒, 𝑓}
Now, let's find the power set by considering all possible combinations:
Empty set: {}
Subset 1: {𝑎, 𝑏}
Subset 2: {𝑐}
Subset 3: {𝑑, 𝑒, 𝑓}
Subset 1 and 2: {𝑎, 𝑏, 𝑐}
Subset 1 and 3: {𝑎, 𝑏, 𝑑, 𝑒, 𝑓}
Subset 2 and 3: {𝑐, 𝑑, 𝑒, 𝑓}
Subset 1, 2, and 3: {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓}
Therefore, the power set of 𝐴 = {{}, {𝑎, 𝑏}, {𝑐}, {𝑑, 𝑒, 𝑓}, {𝑎, 𝑏, 𝑐}, {𝑎, 𝑏, 𝑑, 𝑒, 𝑓}, {𝑐, 𝑑, 𝑒, 𝑓}, {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓}}
To determine the power set of a given set, we need to find all possible subsets of that set.
Let's start with the set A = {{a, b}, {c}, {d, e, f}}.
Step 1: Count the number of elements in the set A. There are three elements: {a, b}, {c}, and {d, e, f}.
Step 2: Determine the number of subsets we can generate. For any set that contains n elements, the number of possible subsets is 2^n.
In our case, we have 3 elements, so the number of possible subsets is 2^3 = 8.
Step 3: Generate all possible subsets.
Subset 1: { }
Subset 2: { {a, b} }
Subset 3: { {c} }
Subset 4: { {d, e, f} }
Subset 5: { {a, b}, {c} }
Subset 6: { {a, b}, {d, e, f} }
Subset 7: { {c}, {d, e, f} }
Subset 8: { {a, b}, {c}, {d, e, f} }
So, the power set of A, denoted as P(A), is {{ }, { {a, b} }, { {c} }, { {d, e, f} }, { {a, b}, {c} }, { {a, b}, {d, e, f} }, { {c}, {d, e, f} }, { {a, b}, {c}, {d, e, f} }}.
To determine the power set of set 𝐴 = {{𝑎, 𝑏},{𝑐},{𝑑, 𝑒, 𝑓}}, follow these steps:
Step 1: List down all the subsets of the given set 𝐴, including the empty set:
Subsets:
∅ (empty set)
{{𝑎, 𝑏}}
{{𝑐}}
{{𝑑, 𝑒, 𝑓}}
{{𝑎, 𝑏}, {𝑐}}
{{𝑎, 𝑏}, {𝑑, 𝑒, 𝑓}}
{{𝑐}, {𝑑, 𝑒, 𝑓}}
{{𝑎, 𝑏}, {𝑐}, {𝑑, 𝑒, 𝑓}}
Step 2: Combine the subsets to form new subsets:
{{𝑎, 𝑏}, {𝑑, 𝑒, 𝑓}, {𝑐}}
{{𝑎, 𝑏}, {𝑐}, {𝑑, 𝑒, 𝑓}, {𝑐}, {𝑑, 𝑒, 𝑓}}
{{𝑎, 𝑏}, {𝑐}, {𝑑, 𝑒, 𝑓}, {𝑐}, {𝑑, 𝑒, 𝑓}, {𝑎, 𝑏}, {𝑑, 𝑒, 𝑓}}
{{𝑎, 𝑏}, {𝑐}, {𝑑, 𝑒, 𝑓}, {𝑐}, {𝑑, 𝑒, 𝑓}, {𝑐}, {𝑎, 𝑏}, {𝑑, 𝑒, 𝑓}}
Step 3: The power set of 𝐴 is the collection of all the subsets obtained in Step 1 and Step 2:
Power set of 𝐴: {∅, {{𝑎, 𝑏}}, {{𝑐}}, {{𝑑, 𝑒, 𝑓}}, {{𝑎, 𝑏}, {𝑐}}, {{𝑎, 𝑏}, {𝑑, 𝑒, 𝑓}}, {{𝑐}, {𝑑, 𝑒, 𝑓}}, {{𝑎, 𝑏}, {𝑐}, {𝑑, 𝑒, 𝑓}}}
So, the power set of 𝐴 is {∅, {{𝑎, 𝑏}}, {{𝑐}}, {{𝑑, 𝑒, 𝑓}}, {{𝑎, 𝑏}, {𝑐}}, {{𝑎, 𝑏}, {𝑑, 𝑒, 𝑓}}, {{𝑐}, {𝑑, 𝑒, 𝑓}}, {{𝑎, 𝑏}, {𝑐}, {𝑑, 𝑒, 𝑓}}}.