1. What is a set

What is a set?

Set

A set is a collection of things, any things, called its elements.

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If $S$ is a set and $x$ is an element in $S$, then $x\in S$. If $x$ is not an element in $S$, then $x\notin S$. If both $x\in S$ and $y\in S$, then $x,y\in S$.

To describe a set:

• We can explicitly name its elements.

  We can form the set $A$ by collecting three things:

  Minecraft, League, ETS2

  Any set defined this way is denoted by listing its elements, separated by commands, surrounded by braces:

  $$A=\{\text{Minecraft, League, ETS2}\}$$

• Anything can be an element of a set.

  A set can be an element of another set.

  For example, the set $B=\{x,\{x,y\}\}$ has two elements:

  • One element is $x$.
  • The other element is the set $\{x,y\}$.

  In this case we can write $x\in B$ and $\{x,y\}\in B$.

Important Features of Sets

The only thing we consider is what is in and isn’t in the set.

• Repeated occurrences don’t matter $\{H,E,L,L,O\}$ and $\{H,E,L,O\}$ describe the same set. It is more ergonomic to only display each once.

• Order of listing doesn’t matter $\{H,E,L,O\}$ and $\{E,H,L,O\}$ describe the same set.

Special sets

Empty set

An empty set is a set with no elements, it is denoted by $\{\}$ or more commonly referred to by the symbol $\boxed \emptyset$.

The empty set has no elements so no matter what $x$ denotes, $x\notin \varnothing$.

Although, an empty set can be an element of another set, $\varnothing \in \{a,\varnothing ,Joe\}$ or $\varnothing \in \{\varnothing \}$.

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Singleton

A singleton is any set with only one element. For example, $\{a\}$ and $\{Monday\}$ are both singletons.

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Equality of sets

Two sets are equal if they contain the same elements.

We can denote this by $A=B$. If two sets are not equal, we write $A\ne B$.