Elementary Set Theory – Set Membership and Inclusion Confusion

discrete mathematicselementary-set-theory

I've recently started studying discrete mathematics for my computer science degree, and I have a doubt regarding membership and inclusion. I'll explain with an example.

Consider the set: $ A = \{1, 2,3,4,5 \} $

True or false?

  1. $1 \in A$

  2. $\{1\} \in A$

  3. $1 \subset A$

  4. $\{1\} \subset A$

Number one is obviously true, because the element $1$ is a member of $A$.

Number two, as far as I understand, should be false because it makes no sense. Membership is between a set and an element, not between two sets. Correct?

Number three: I'm not sure what the answer is. Can the concept of inclusion be applied between an element and a set or only between sets?

Number four is obviously true, because the singleton set $\{1\}$ is a subset of $A$.

  • Are my answers correct?
  • Can the concept of membership be applied between two sets? Or is it only always between an element and a set?
  • Can the concept of inclusion be applied between an element and a set? Or is it only always between two sets?
  • Consider $B = \{1, 2, 3\}, C = \{\{1\}, 2, 3\}$.
    • Is $B=C$ true? (Can a singleton set be considered as an element?)
    • Is $1 \in C$ true?
    • Is $\{1\} \in C$ true?
    • Is $1 \subset C$ true?
    • Is $\{1\} \subset C$ true?

Best Answer

You're correct that 1 and 4 are true.

On the other hand, if we're doing Zermelo-Fraenkel set theory, then the truthvalues of 2 and 3 cannot be determined from the information given. For example, 2 says that $\{1\} \in A$, or in other words that $\{1\} = 1$ or $\{1\} = 2$ or $\{1\} = 3$ or $\{1\} = 4$ or $\{1\} = 5$. Are any of those statements true? Well they could be. In Zermelo's approach to defining the natural numbers, $\{1\} = 2$ is true, indeed this is the definition of $2$. However, in von Neumann's approach, the set $\{1\}$ is not a natural number and thus not an element of $A$.

Similarly, the truthvalue of 3 cannot be determined. It all comes down to how you define the notion "natural number."

On the other hand, other approaches to the foundations would deem 2 and 3 "ill-formed" and therefore nonsensical. They're not even false; they're just nonsense.

To answer your questions: Membership can be a relation between two sets (at least in ZFC). For example, $\mathbb{R} \in \{\mathbb{R}\}$ is true. Furthermore, there is no distinction between "elements" and "sets" in ZFC; everything is a set, and "element" is a relationship: we can say the set $\mathbb{R}$ is an element of $\{\mathbb{R}\}$, but there's no point in saying that something "is" an element, period.

This was quite a rushed answer because I need to get back to my assignments, so please comment with any questions you may have.

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