Unordered Triples

Why

$\set{a} \cup \set{b} = \set{a, b}$

Definition

Let $a$, $b$ and $c$ denote objects. From the associativity of pair unions (seePair Unions), we have

\[ (\set{a} \cup \set{b}) \cup \set{b} = \set{a} \cup (\set{b} \cup \set{c}). \]

So we will drop the parentheses, and write $\set{a} \cup \set{b} \cup \set{c}$. We call such a set the unordered triple of $a$, $b$ and $c$. The unordered triple of $a$, $b$ and $c$ is the set containing these elements and no others.

Notation

Such sets are so commonplace that we denote the unordered triple of $a$, $b$ and $c$ by $\set{a, b, c}$.

Quadruples

Let $d$ denote an object. Again, the associativity of pair unions allows us to drop the parentheses from

\[ (((\set{a} \cup \set{b}) \cup \set{c}) \cup \set{d})). \]

We can therefore write $\set{a} \cup \set{b} \cup \set{c} \cup \set{d}$ without ambiguity. We call this set the unordered quadruple. As before, the unordered quadruple contains of $a$, $b$, $c$ and $d$ contains $a$, $b$, $c$, and $d$ and nothing besides these.

Notation

We denote the unordered quadruple of the objected denoted by $a$, $b$, $c$ and $d$, denote this set by $\set{a, b, c, d}$.

The case of several named objects

In a similar way we speak of unordered pentuples, unordered sextuples, unordered septuples and so on. If we have several objects named, we denote the set containing these objects be writing their names in between the left brace $\{$ and right brace$\}$, separating the names by commas. For example, if we $A$, $b$, $x$ and $Y$ and $z$ denote objects, then we denote the set containing these elements by

\[ \set{A, b, x, Y, z}. \]