Question

Determine whether each of these sets is the power set of a set, where a and b are distinct elements.

1. {0}

2. {0, { a } , { 0, a }}

3. {0, { a } }

4. {0, { a }, { b }, { a, b }}

Answer 1

$1:\\\left\{ \emptyset \right\} =2^{\emptyset}\,\,-\,\,yes\\2:\\\left\{ \emptyset ,\left\{ a \right\} ,\left\{ \emptyset ,a \right\} \right\} -no, the\,\,power\,\,set\,\,contains\,\,2^n\,\,elements\\3:\\\left\{ \emptyset ,\left\{ a \right\} \right\} =2^{\left\{ a \right\}}-yes\\4:\\\left\{ \emptyset ,\left\{ a \right\} ,\left\{ b \right\} ,\left\{ a,b \right\} \right\} =2^{\left\{ a,b \right\}}-yes$