Cardinality of set-Discrete math
Take $A=\{1,2,3\}$ and $B=\{1,2,5\}$.
If we unionized them together it would be $A\cup B=\{1,2,3,5\}$ and if we intersected them it would be $A\cap B=\{1,2\}$.
However, if we change $B$ to $B=\{\{1,2\},2,5\}$ would a set within a set change the previous answers and why?
$\endgroup$ 31 Answer
$\begingroup$Yes, it changes the union and intersection. We now have
$$A\cup B=\{1,2,3\}\cup\big\{\{1,2\},2,5\big\}=\big\{1,2,3,\{1,2\},5\big\}$$
and
$$A\cap B=\{1,2,3\}\cap\big\{\{1,2\},2,5\big\}=\{2\}\;.$$
The set $\{1,2\}$ is an object in its own right, distinct from the objects $1$ and $2$; $1$ and $2$ are elements of $A$, and $\{1,2\}$ is now an element of $B$, so these three distinct objects are all elements of the union $A\cup B$, along with the other elements of $A$ and $B$.
$1$ is now an element of an element of $B$, but it is not itself an element of $B$. Thus, the only object that $A$ and $B$ now have in common is $2$, and the intersection is therefore $\{2\}$.
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