Definition:Image of Class under Mapping/Warning

Image of Class under Mapping: Warning

Let $f: V \to V$ be a mapping.

Let $A$ be a class.

Let $x \in A$ be a set of sets

Then:

$f \sqbrk x$ is not necessarily the same as $\map f x$

where:

$f \sqbrk x$ denotes the image of $x$ under $f$ where $x$ is treated as a class

$\map f x$ denotes the image of $x$ under $f$ where $x$ is treated as an element of the class $A$.

Thus:

$\map f x$ is what you get by applying $f$ to $x$

$f \sqbrk x$ is what you get by applying $f$ to each of the elements of $x$ (but not $x$ itself) and then gathering the results into a set.

Consider the mapping $f$ defined on the class of all ordinals $\On$ as:

$\forall \alpha \in \On: \map f \alpha = 5 \times \alpha$

where $\times$ denotes ordinal multiplication.

Consider the ordinal $4$

We have that:

$\map f 4 = 20$

but:

$f \sqbrk 4 = \set {5 \times \beta: \beta \in 4}$

$4 = \set {0, 1, 2, 3}$

and so:

$f \sqbrk 4 = \set {0, 5, 10, 15}$

As can be seen, they are not the same thing at all.

$\blacksquare$

2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 1$ A few preliminaries