2. Determine the power set of 𝐴𝐴 = {{𝑎𝑎, 𝑏𝑏},{𝑐𝑐},{𝑑𝑑, 𝑒𝑒, 𝑓𝑓}}.
The power set of a set is the set of all possible subsets of that set, including the empty set and the set itself.
The set 𝐴 = {{𝑎𝑎, 𝑏𝑏},{𝑐𝑐},{𝑑𝑑, 𝑒𝑒, 𝑓𝑓}} has a total of 3 elements: {𝑎𝑎, 𝑏𝑏}, {𝑐𝑐}, and {𝑑𝑑, 𝑒𝑒, 𝑓𝑓}.
To determine the power set, we need to list all possible subsets of these elements:
- The empty set: {}
- Subsets with only one element: {𝑎𝑎, 𝑏𝑏}, {𝑐𝑐}, {𝑑𝑑, 𝑒𝑒, 𝑓𝑓}
- Subsets with two elements: {𝑎𝑎, 𝑏𝑏}, {𝑐𝑐}, {𝑑𝑑, 𝑒𝑒, 𝑓𝑓}, {𝑎𝑎, 𝑏𝑏}, {𝑎𝑎, 𝑐𝑐}, {𝑏𝑏, 𝑐𝑐}
- Subsets with three elements: {𝑎𝑎, 𝑏𝑏, 𝑐𝑐}, {𝑎𝑎, 𝑏𝑏, 𝑑𝑑, 𝑒𝑒, 𝑓𝑓}, {𝑐𝑐, 𝑑𝑑, 𝑒𝑒, 𝑓𝑓}
Therefore, the power set of 𝐴𝐴 is:
{{}, {𝑎𝑎, 𝑏𝑏}, {𝑐𝑐}, {𝑑𝑑, 𝑒𝑒, 𝑓𝑓}, {𝑎𝑎, 𝑏𝑏}, {𝑎𝑎, 𝑐𝑐}, {𝑏𝑏, 𝑐𝑐}, {𝑎𝑎, 𝑏𝑏, 𝑐𝑐}, {𝑎𝑎, 𝑏𝑏, 𝑑𝑑, 𝑒𝑒, 𝑓𝑓}, {𝑐𝑐, 𝑑𝑑, 𝑒𝑒, 𝑓𝑓}}
To determine the power set of a set, you need to find all possible subsets of that set, including the empty set and the set itself.
Given set A = {{aa, bb}, {cc}, {dd, ee, ff}}, let's find its power set step-by-step:
Step 1: List down all the individual elements of the set.
{aa, bb}, {cc}, {dd, ee, ff}
Step 2: Find all possible combinations of subsets.
{}, // empty set
{{aa, bb}}, // one subset containing {aa, bb}
{{cc}}, // one subset containing {cc}
{{dd, ee, ff}}, // one subset containing {dd, ee, ff}
{{aa, bb}, {cc}}, // two subsets {aa, bb} and {cc}
{{aa, bb}, {dd, ee, ff}}, // two subsets {aa, bb} and {dd, ee, ff}
{{cc}, {dd, ee, ff}}, // two subsets {cc} and {dd, ee, ff}
{{aa, bb}, {cc}, {dd, ee, ff}} // three subsets {aa, bb}, {cc}, and {dd, ee, ff}
Therefore, the power set of A, denoted as P(A), is:
P(A) = { {}, {aa, bb}, {cc}, {dd, ee, ff}, {aa, bb, cc}, {aa, bb, dd, ee, ff}, {cc, dd, ee, ff}, {aa, bb, cc, dd, ee, ff} }
To determine the power set of a set, you need to find all possible subsets of that set. Here's how you can find the power set of set A = {{a, b},{c},{d, e, f}}:
1. Write down the empty set {} as the first element of the power set.
2. Start with the first element of set A, which is {a, b}. It can either be included or excluded in each subset.
a. Include {a, b} in the first subset: {{a, b}}
b. Exclude {a, b} in the second subset: {}
3. Move to the next element of set A, which is {c}. It can either be included or excluded in each subset.
a. Include {c} in each existing subset: {{a, b}, {c}}
b. Leave each existing subset as is: {{a, b}, {}}
4. Move to the next element of set A, which is {d, e, f}. It can either be included or excluded in each subset.
a. Include {d, e, f} in each existing subset: {{a, b}, {c}, {d, e, f}}
b. Leave each existing subset as is: {{a, b}, {c}, {}}
5. Combine all the generated subsets to get 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}}
Therefore, 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}}.