Universal Set and Subsets

Defn. Set

• A set is a well-defined collection of objects.
• These objects are called elements or members of the set.
• The collection of all objects of interest is called the universal set or universe.

Note.

• Notations: Uppercase letters for sets, e.g. $A,B,C$ ; $u$ or $\Omega$ for universals.
• $x\in A$ is read as ”$x$ in $A$” or ”$x$ is an element of $A$”
• $A=3k|\mathbb{Z}$ is read as ”$A$ is the set of numbers of form $3k$ such that $k$ is an integer.”

Examples

• Finite set: $\{1,2,3\}$, $\{2,4,6,8,\dots ,1000\}$
• Countably Infinite set: $\mathbb{Z}$, $\mathbb{Q}$, $2,4,6,8,\dots$
• Uncountable set: $(0,1)$, $(-\infty ,1]$, $[100,\infty )$

Roster Method/Listing Method	Set-Builder Notation/Rule Method
${\mathbb{Z}}_E=\{\dots ,-2,0,2,4,\dots \}$	${\mathbb{Z}}_0=\{n:n=2k$ for some $k\in \mathbb{Z}\}$
${\mathbb{Z}}^{-}=\{\dots ,-4,-3,-2,-1\}$	$[1,\infty )=/x\in \mathbb{R}:x\ge \frac{1}{}$
Set of primes less than 3 = $\{2\}$	${\mathbb{Q}}^{+}=\{x\in \mathbb{R}:x=\frac{p}{q}$ where $p,q\in \mathbb{Z}\}$
$\mathbb{N}=\{1,2,3,4,5,\dots \}$	$\mathbb{C}=\{a+bi:a,b\in \mathbb{R}\wedge i=\sqrt{-1}\}$
Set of multiples of 5 = $\{\dots ,-10,-5,0,5,10,\dots \}$	Set of multiples of 3 = $\{n:3\Vert n$ for some $n\in z$

Defn. Cardinality

• The cardinality or size of a finite set is the number of elements it has.
• Notations: $|A|$ or $n(A)$

Defn. Empty and Singleton Sets

• The empty set is the set that has no elements. Written as $\varnothing$.
• A singleton set is the set that has only one element.

Note

• $\varnothing$ is a finite set with $|\varnothing |=0$
• $\varnothing$ is not the same as $\{\varnothing \}$. $|\{\varnothing =1\}|$.

Defn. Subset

Let $A$ and $B$ be sets.

• $A$ is a subset of $B$ if every element of $A$ also belongs to $B$. $A\subseteq B⟺\forall x,(x\in A⟹x\in B)$ or $B\supseteq A$.
• $A$ is a proper subset of $B$ if every element of $A$ also belongs to $B$. $A\subset B⟺\forall x,(x\in A⟹x\in B)\wedge \exists x,(x\in B\wedge n\notin A$)
• $A$ and $B$ are equal if they contain exactly the same elements.
  • $A=B⟺A\subseteq B\wedge B\subseteq A$
  • $A=B⟺\forall x,(x\in Ax\in B)\wedge \forall x,(x\in B=x\in A)$
  • $A=B⟺\forall x,(x\in A⟺x\in B)$

Examples

• $\mathbb{Q}\subseteq \mathbb{R}$
• $\{1,2\}\subseteq (0,1)$
• ${\mathbb{Z}}_E\subset \mathbb{Z}$
• $\mathbb{Z}\subset \mathbb{R}$
• ${\mathbb{Z}}^{+}=\mathbb{N}$
• ${\mathbb{Z}}_E=\{x\in \mathbb{Z}:2|x\}$
• $\{1\}⊈[10,\infty )$
• $\mathbb{Q}\subset {\mathbb{Q}}^C$
• $\{a\}\not\subset \{a\}$
• $\{x,y,z\}\not\subset \{x,y,z\}$

Thm.

1. $\varnothing \subseteq A$ for any set $A$.
2. $A\subseteq A$ for any set $A$.
3. If $A\subseteq B$ and $B\subseteq C$ then $A\subseteq C$ for any sets $A,B,C$.

Cor.

There is only one set with no elements ($\varnothing$).