HW1 Problem 5

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Problem 5

List the elements of the power set of the given set and give the cardinality of the power set.

(a) $\varnothing$ (b) $\{a\}$ (c) $\{a,b\}$ (d) $\{a,b,c\}$

Solution

(a) $\mathscr{P}(\varnothing)$ = $\{\varnothing\}$
$|\{\varnothing\}|$ = 1

(b) $\mathscr{P}(\{a\})$ = $\{\varnothing, \{a\}\}$
$|\{\varnothing, \{a\}\}|$ = 2

(c) $\mathscr{P}(\{a,b\})$ = $\{\varnothing, \{a\}, \{b\}, \{a,b\}\}$
$|\{\varnothing, \{a\}, \{b\}, \{a,b\}\}|$ = 4

(d) $\mathscr{P}(\{a,b,c\})$ = $\{\varnothing, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}\}$
$|\{\varnothing, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}\}|$ = 8