[Math] Understanding basic sets/subsets in discrete math by determining if statements are true or false

discrete mathematicselementary-set-theory

Hi I'm working on some basic set/subset comparison statements in the form of True/False. There are 4 statements that I have to determine whether they're true or false. I think I understand the first 3, but not sure about the last.

  1. x $\in$ {x}: True. x is an element in the set x?

  2. x $\in$ {{x}}: False. x is the subset of the set?

  3. {x} $\in$ {x}: False. element x is in element x?

  4. {x} ⊆ {{x}}: True? Not sure why lol

Any suggestions/corrections are greatly appreciated. Thanks!

Best Answer

You are correct for the first three, but not the fourth. To say that

$$A \subseteq B$$

means that every element of $A$ is also an element of $B$. The only element of $\{x\}$ is $x$ itself, and the only element of $B$ is $\{x\}$. These aren't the same, so the statement

$$\{x\} \subseteq \{\{x\}\}$$ is false.

In short, $x \notin \{\{x\}\}$, since the only thing in $\{\{x\}\}$ is $\{x\}$.

Related Question