Example 1.7 $$ \mathcal{P}(\{a, b\})= $$ $$ \{\emptyset,\{a\},\{b\},\{a, b\}\} $$ $$ \square $$ Sometimes $$ \mathcal{P}(S) $$ is written $$ 2^{S} $$ $$ \square $$ For a finite set $$ S $$ $$ |\mathcal{P}(S)|=2^{|S|} $$

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This example demonstrates the concept of a power set.

1. The power set of any set S, denoted by ℘(S) or 2^S, is the set of all subsets of S.  
2. For the set S = {a, b}, its subsets are:
 • The empty set: ∅  
 • The singleton set: {a}  
 • The singleton set: {b}  
 • The set itself: {a, b}  
So, ℘({a, b}) = {∅, {a}, {b}, {a, b}}.

3. An important fact about power sets of finite sets is that if a set S has |S| elements, then the power set ℘(S) has 2^(|S|) elements. In this example, |{a, b}| = 2, so |℘({a, b})| = 2² = 4.

This completes the explanation for Example 1.7.
The power set of {a, b} is {∅, {a}, {b}, {a, b}}. For any finite set, the number of subsets is $$2$$ raised to the number of elements in the set.

Example 1.7 \( \mathcal{P}(\{a, b\})= \) \( \{\emptyset,\{a\},\{b\},\{a, b\}\} \) \( \square \) Sometimes \( \mathcal{P}(S) \) is written \( 2^{S} \) \( \square \) For a finite set \( S \) \( |\mathcal{P}(S)|=2^{|S|} \)

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