# First Projection on Ordered Pair of Sets

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## 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

- 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.6$: Functions: Exercise $1.6.6 \ \text{(i)}$