Empty function

In mathematics, an empty function is a function whose domain is the empty set $\emptyset$ . For each set $A$ , there is exactly one such empty function

The graph of an empty function is a subset of the Cartesian product $\emptyset \times A$ . Since the product is empty the only such subset is the empty set $\emptyset$ . The empty subset is a valid graph since for every $x$ in the domain $\emptyset$ there is a unique $y$ in the codomain $A$ such that $(x, y) \in \emptyset \times A$ . This statement is an example of a vacuous truth since "there is no $x$ in the domain."

单词	Empty function
释义	Empty function 英语百科 Empty function In mathematics, an empty function is a function whose domain is the empty set $\emptyset$ . For each set $A$ , there is exactly one such empty function The graph of an empty function is a subset of the Cartesian product $\emptyset \times A$ . Since the product is empty the only such subset is the empty set $\emptyset$ . The empty subset is a valid graph since for every $x$ in the domain $\emptyset$ there is a unique $y$ in the codomain $A$ such that $(x, y) \in \emptyset \times A$ . This statement is an example of a vacuous truth since "there is no $x$ in the domain."
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