Definition:One-to-Many Relation
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Definition
A relation $\RR \subseteq S \times T$ is one-to-many if and only if:
- $\RR \subseteq S \times T: \forall y \in \Img \RR: \tuple {x_1, y} \in \RR \land \tuple {x_2, y} \in \RR \implies x_1 = x_2$
That is, every element of the image of $\RR$ is related to by exactly one element of its domain.
Note that the condition on $t$ concerns the elements in the image, not the codomain.
Thus a one-to-many relation may leave some element(s) of the codomain unrelated.
Examples
Inverse Sine
Let $\RR \subseteq \R \times \R$ be the relation on $\R$ defined as:
- $\forall \tuple {x, y} \in \R \times \R: \tuple {x, y} \in \RR \iff x = \sin y$
Then $\RR$ is a one-to-many relation.
Also known as
A one-to-many relation is also known as:
- a one-many relation
- an injective relation
- a left-unique relation
- a single-rooted relation
- a one-many correspondence
Also see
- Results about one-to-many relations can be found here.
Sources
- 1939: E.G. Phillips: A Course of Analysis (2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Number: $1.2$ Fundamental notions
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): many-one correspondence
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): one-to-many