First Projection on Ordered Pair of Sets

From ProofWiki
Jump to navigation Jump to search

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

$\blacksquare$


Sources