What is the cardinality of the set A,B where
My answer was 4 for set A and 6 for set B. When i looked at the text book for answer it is not provided in the text book and now i'm wondering what the correct answer would be.
A = | { { a, b, c }, { b, c, a }, { c, b, a }, { b, a, c } } |
B = | { { 1, 2, 3, { a, b, c } }, { 2, 3, 4 { b, c, d } }, { 3, 4, 5 { c, d, e } } } |
Thanks
$\endgroup$ 42 Answers
$\begingroup$The cardinality of $A$ relies on two fundamental concepts about all sets:
- Sets do not contain duplicates, so $\{x, x\}$ is always the same as $\{x\}$
- Two sets are equal if and only if they contain the same elements. This means that $\{a,b,c\} = \{c,b,a\}$ and so on.
So while $A$ has been written to look like it contains 4 sets, all 4 of those sets contain the same elements and are actually all the same set. And because sets don't contain duplicates. $A$ should really be written as $\{\{a,b,c\}\}$, so its cardinality is 1.
The cardinality of $B$ is 3. When calculating the cardinality of a set, you don't "look into" the members of the set and count nested sets. So we can rewrite $B$ as such:
- $X = \{ 1, 2, 3, \{ a, b, c \} \}$
- $Y = \{ 2, 3, 4, \{ b, c, d \} \}$
- $Z = \{ 3, 4, 5, \{ c, d, e \} \}$
- $B = \{X,Y,Z\}$
A set is defined by its members, not by the order in which they are listed: $\{a,b,c\}$, $\{b,c,a\}$, $\{c,b,a\}$, and $\{b,a,c\}$ are just four different names for the same set, the one whose members are $a,b$, and $c$. Thus, $A=1$: that set has only one distinct member.
$B=3$, because the set in question has the members $\big\{\{1,2,3,\{a,b,c\}\big\}$, $\big\{\{2,3,4,\{b,c,d\}\big\}$, and $\big\{\{3,4,5,\{c,d,e\}\big\}$. Each of these three members is itself a set with four members, one of which is a set with three members.
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