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If B = {a, b, c, d}, what is the power set of B?

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Question: If B = {a, b, c, d}, what is the power set of B?

Options:

  1. {∅, {a}, {b}, {c}, {d}}
  2. {∅, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}}
  3. {∅, {a, b}, {c, d}}
  4. {∅, {a, b, c}, {d}}

Correct Answer: {∅, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}}

Solution: 

The power set of a set with n elements has 2^n subsets. Here, n = 4, so the power set has 2^4 = 16 subsets.

If B = {a, b, c, d}, what is the power set of B?

Practice Questions

Q1
If B = {a, b, c, d}, what is the power set of B?
  1. {∅, {a}, {b}, {c}, {d}}
  2. {∅, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}}
  3. {∅, {a, b}, {c, d}}
  4. {∅, {a, b, c}, {d}}

Questions & Step-by-Step Solutions

If B = {a, b, c, d}, what is the power set of B?
Correct Answer: { {}, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d} }
  • Step 1: Identify the set B. In this case, B = {a, b, c, d}.
  • Step 2: Count the number of elements in set B. There are 4 elements: a, b, c, and d.
  • Step 3: Use the formula for the power set, which is 2^n, where n is the number of elements in the set.
  • Step 4: Calculate 2^4, since n = 4. This equals 16.
  • Step 5: Understand that the power set includes all possible subsets of B, including the empty set and the set itself.
  • Step 6: List all the subsets: {}, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}.
  • Step 7: Confirm that there are 16 subsets listed, which matches the calculation from Step 4.
  • Power Set – The power set of a set is the set of all possible subsets, including the empty set and the set itself.
  • Exponential Growth of Subsets – The number of subsets of a set grows exponentially with the number of elements, specifically 2^n for a set with n elements.
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