Determine whether each of these pairs of sets are equal$$A = \{\{1\}\} \qquad \qquad B = \{1, \{1\}\}$$
I believe $A$ is equal to $B$ because all elements in $A$ are in $B$, but the answer says that it's not.
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Sign up to join this communityDetermine whether each of these pairs of sets are equal$$A = \{\{1\}\} \qquad \qquad B = \{1, \{1\}\}$$
I believe $A$ is equal to $B$ because all elements in $A$ are in $B$, but the answer says that it's not.
Think of $A$ as a bag which contains within it another smaller bag with a one in it.
$A=\underbrace{\{~~~~~~~\overbrace{\{1\}}^{\text{second bag}}~~~~~~~~\}}_{\text{first bag}}$
On the other hand, $B$ is a bag which contains in it not only a second bag with a one in it, but also a one which is loose.
$B=\underbrace{\{~~~~~~~~\overbrace{\{1\}}^{\text{second bag}}~~~~~\overbrace{1}^{\text{this too}}~~~~~~~\}}_{\text{first bag}}$
$1\in B$ but $1\not\in A$. There is no "loose 1" in $A$, there is only a bag with a one in it in $A$.
Thus, $A\neq B$
You're correct that all elements in $A$ are in $B$, but not the other way around - $B$ includes the element $1$, but $A$ only has $\{1\}$. Think of it like boxes - $B$ is a box that includes one item and also a box that itself contains one item; $A$ is just a box containing a box containing an item.