Image of a function ($f(X)$)

For a function $f:X\to Y$ and a subset $E\subseteq X$, its image isThe full image of a function is $f(X)$. The set notation ignores repeated outputs: in the finite example above, $f(\{a,b\})=\{0\}$, rather than a list containing zero twice. A function collision can therefore reduce the cardinality of an image.

Images of a function preserve unions:An output reached from the union is reached from at least one of its two input sets. For intersections, however, onlyholds in general. In the finite example, take $E=\{a\}$ and $F=\{b\}$. Their intersection is the empty set, so $f(E\cap F)=\varnothing$. But $f(E)=f(F)=\{0\}$, giving $f(E)\cap f(F)=\{0\}$. Distinct witnesses in the two input sets can produce the same output.

For an injective function, an output shared by $f(E)$ and $f(F)$ must come from the same input. That input lies in $E\cap F$, so equality holds. Conversely, preserving intersections for every pair of input subsets forces injectivity: two distinct inputs with the same output would give the singleton counterexample above. The image and preimage laws are stated in Jiří Lebl, Basic Analysis I, Propositions 0.3.15–0.3.16; the finite example makes the missing reverse inclusion visible.