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I am reading an engineering mathematics book whose author says that the term mapping and function have the same meaning and whether one calls $f:X\rightarrow Z$ a function or mapping is a matter of choice. Nevertheless, I do remember from calculus course that the two terms cannot be used interchangeably and I think that mapping is a more general term than function. So, what am I missing here? Is the author right?

J. W. Tanner
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Dimitris
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    "...is a more general term than function." A more general term for a function would be a relation. A function is just a relation with more specific requirements. If you are looking for something like a function but has multiple outputs for a single input or has some inputs who have no outputs, the term you want is "relation"... not mapping. – JMoravitz Jan 06 '21 at 14:32

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The author is correct. What you're thinking of is a relation, which is defined as any subset of a Cartesian product of two sets $A \times B = \{(a,b) : a \in A, b \in B\}$. The relation is then true for a pair $a,b$ if $(a,b)$ is an element of the subset. The difference between this and a function/mapping is that with a function $f$, for every $a \in A$ there must be exactly one corresponding element $(a,b) \in f$. This in turn means that a function maps exactly one $b\in B$ to every $a \in A$ and we write $f(a)=b$.

Kenny
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