Preimage of Element under Projection
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Theorem
Let $A$ and $B$ be sets.
Let $A \times B$ be the cartesian product of $A$ and $B$.
Let $\pr_1: A \times B \to A$ be the first projection of $A \times B$.
Let $a \in A$.
Then:
- $\pr_1^{-1} \sqbrk {\set a} = \set {\tuple {a, b}: b \in B}$
that is:
- $\pr_1^{-1} \sqbrk {\set a} = \set a \times B$
Proof
Directly apparent from the definition of cartesian product.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 6$: Functions: Exercise $5$