Question

What is the number of Power set of A show step by step? A = {a,b, {a}, 3, {3}}.​

Answer 1

By "number of Power set of A" I wonder if you mean the cardinality of the power set.

For any finite set with $n$ elements, its power set contains $2^n$ elements.

Here $|A| = 5$, so the power set will contain $|\mathcal P(A)| = 2^5 = \boxed{32}$ elements.

Recall that the power set is the set of all subsets of a given set

• subsets of $A$ of size 0 (1):

$\emptyset$

• subsets of size 1 (5):

$\{a\}\\ \{b\}\\ \{\{a\}\}\\ \{3\}\\ \{\{3\}\}$

• subsets of size 2 (10):

$\{a,b\}\\ \{a,\{a\}\}\\ \{a, 3\}\\ \{a, \{3\}\}\\ \{b,\{a\}\}\\ \{b,3\}\\ \{b,\{3\}\}\\ \{\{a\},3\}\\ \{\{a\}, \{3\}\}\\ \{3,\{3\}\}$

• subsets of size 3 (10):

$\{a,b,\{a\}\}\\ \{a,b,3\}\\ \{a,b,\{3\}\}\\ \{a,\{a\},3\}\\ \{a,\{a\},\{3\}\} \\ \{a,3,\{3\}\}\\ \{b,\{a\},3\}\\ \{b,\{a\},\{3\}\}\\ \{\{a\},3,\{3\}\}$

• subsets of size 4 (5):

$\{a,b,\{a\},3\}\\ \{a,b\{a\},\{3\}\}\\ \{a,b,3,\{3\}\}\\ \{a,\{a\},3,\{3\}\}\\ \{b,\{a\},3,\{3\}\}$

• subsets of size 5 (1):

$\{a,b,\{a\},3,\{3\}\}$