Injective vs Surjective vs Bijective

With relation (pun lol) to functions, you can have a whole lot.

Only relations that are injective and/or surjective are Functions. What is that you may be asking?

Injective (One-to-one):

• A function $f:A\to B$ if every $A$ has a unique (i.e. not repeated) image in $B$. I.E. No 2 distinct elements map onto B.
• Not all B are necessarily mapped. Some are lonely outputs waiting for an input to connect to it.
• I.E. No “doubling up.” Different inputs give different outputs.

Surjective (Onto):

• A function $f:A\to B$ if every $B$ is mapped onto by at least a single element of $A$.
• No “gaps” in the output. The entire codomain is covered.

Bijective (One-To-One and Onto - AKA Perfectly):

• The most perfect function $f$, as it’s every A perfectly mapping to every B, with no overlap and no duplicates.
• These types of functions have an inverse, $f^{-1}$. This “undoes” whatever $f$ did.