Question

Find the power set of each of these sets, where a and b are distinct elements. a) {a} b) {a, b} c) {1, 2, 3, 4} show steps

Answer 1

a) {{}, {a}}.

b) {{}, {a}, {b}, {a, b}}.

c) {{}, {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}, {4}, {1, 4}, {2, 4}, {1, 2, 4}, {3, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}.

Explanation:

The power set of a set is the set of all subset of the set in question. The number of power sets (including the empty set) of a set with
`n` (where
`n \in \mathbb{Z}`) unique elements is
`2^(n)`.

In other words, there shall be

• 2 items in the power set of a),
• 4 items in the power set of b), and
• 16 items in the power set of c).

This explanation shows how to find the power set using binary numbers (only 0 and 1.) (Credit: Mathsisfun.)

a)

List all the binary numbers that are equivalent to decimals ranging from 0 to
`2 - 1 = 1`.

`\begin{array}l\text{Decimal}&\text{Binary}\\ 0 & 0 \\ 1 & 1\end{array}`.

Reverse the original set. Each digit in the binary number corresponds to a member of the original set (i.e. a letter in a) and b) or a number in c).) 0 means that the element is absent in the subset and 1 means that the element is present.

`\begin{array}la & \text{Element of the Power Set}\\ 0 & \{\}\\ 1 & \{a\}\end{array}`.

The power set of a) thus contains:

• {} and
• {a}.

b)

Similarly, list all the binary numbers that are equivalent to decimals ranging from 0 to
`4 - 1 = 3`.

`\begin{array}l\text{Decimal}&\text{Binary}\\ 0 & 00 \\ 1 & 01 \\ 2 & 10 \\ 3 & 11\end{array}`.

`\begin{array}ccb & a & \text{Element of the Power Set}\\ 0 & 0 & \{\}\\ 0 & 1 & \{a\}\\ 1 & 0 & \{b\} \\ 1 & 1 & \{a, b\}\end{array}`.

The power set of b) thus contains:

• {},
• {a},
• {b}, and
• {a, b}.

c)

Similarly, list all the binary numbers that are equivalent to decimals ranging from 0 to
`16 - 1 = 15`.

`\begin{array}l\text{Decimal}&\text{Binary}\\ 0 & 0000 \\ 1 & 0001 \\ 2 & 0010 \\ 3 & 0011\\4 & 0100 \\ 5 & 0101\\ 6 & 0110 \\ 7 & 0111\\ 8 & 1000\\ 9 & 1001\\ 10 & 1010\\ 11 & 1011\\ 12& 1100 \\13 & 1101 \\ 14 & 1110\\ 15 & 1111 \end{array}`.

`\begin{array}cccc4 & 3 & 2 &1& \text{Element of the Power Set}\\ 0 & 0 & 0 & 0 &\{\}\\ 0 & 0 & 0 & 1 & \{1\}\\ 0 & 0 & 1 & 0 & \{2\} \\ 0 & 0 &1 & 1 & \{1, 2\} \\ 0 & 1 & 0 & 0 & \{3\} \\ 0 & 1 & 0 & 1& \{1, 3\}\\ 0 & 1 & 1 & 0& \{2, 3\}\\ 0 & 1 & 1 & 1 & \{1, 2, 3\} \\ 1 & 0 & 0 & 0 & \{4\} \\ 1 & 0 & 0 & 1 & \{1, 4\}\\ 1& 0 & 1 &0&\{2, 4\}\\ 1 & 0 & 1 & 1 &\{1, 2, 4\}\\ 1 & 1 & 0 & 0 & \{3, 4\} \\ 1 & 1 & 0 & 1 & \{1, 3, 4\} \\ 1 & 1 & 1 & 0 & \{2, 3, 4\} \\ 1 & 1 & 1 & 1 & \{1, 2, 3, 4\}\end{array}`.

The power set of c) thus contains:

• {},
• {1},
• {2},
• {1, 2},
• {3},
• {1, 3},
• {2, 3},
• {1, 2, 3}
• {4},
• {1, 4},
• {2, 4},
• {1, 2, 4},
• {3, 4},
• {1, 3, 4},
• {2, 3, 4}, and
• {1, 2, 3, 4}.