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:
- $\ds \map {\pr_1} w = \bigcup \bigcap w$
where $\ds \bigcup$ and $\ds \bigcap$ 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:
\(\ds \bigcup \bigcap w\) | \(=\) | \(\ds \bigcup \bigcap \tuple {a, b}\) | Definition of $w$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcup \bigcap \set {\set a, \set {a, b} }\) | Definition of Ordered Pair | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \bigcup {\set a \cap \set {a, b} }\) | Intersection of Doubleton | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcup \set a\) | Definition of Set Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds a\) | 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 $1.6.6 \ \text{(i)}$