Equality of Mappings/Unequal/Examples/Exponential Functions

Example of Mappings which are Unequal

Let $\theta: \R \to \R$ be the mapping from the set of real numbers to itself defined as:

$\forall x \in \R: \map \theta x := e^x$

where $e^x$ denotes the real exponential function.

Let $\phi: \R \to \R_{>0}$ be the mapping from the set of real numbers to the set of (strictly) positive real numbers defined as:

$\forall x \in \R: \map \phi x := e^x$

Then:

$\phi \ne \theta$

because, while $\Dom \theta = \Dom \phi$, and $\forall x \in \R: \map \theta x = \map \phi x$, they have different codomains:

$\Cdm \theta = \R \ne \R_{>0} = \Cdm \phi$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.2$. Equality of mappings: Example $46$