# First Projection on Ordered Pair of Sets

## Theorem

Let $a$ and $b$ be sets.

Let $w = \tuple {a, b}$ denote the ordered pair of $a$ and $b$.

Let $\map {\pr_1} w$ denote the first projection on $w$.

Then:

$\displaystyle \map {\pr_1} w = \bigcup \bigcap w$

where $\displaystyle \bigcup$ and $\displaystyle \bigcap$ denote union and intersection respectively.

## Proof

We have by definition of first projection that:

$\map {\pr_1} w = \map {\pr_1} {a, b} = a$

Then:

 $\displaystyle \bigcup \bigcap w$ $=$ $\displaystyle \bigcup \bigcap \tuple {a, b}$ Definition of $w$ $\displaystyle$ $=$ $\displaystyle \bigcup \bigcap \set {\set a, \set {a, b} }$ Definition of Ordered Pair $\displaystyle$ $=$ $\displaystyle \map \bigcup {\set a \cap \set {a, b} }$ Intersection of Doubleton $\displaystyle$ $=$ $\displaystyle \bigcup \set a$ Definition of Set Intersection $\displaystyle$ $=$ $\displaystyle a$ Union of Singleton

$\blacksquare$