# Notation for the set of all injections from $A$ into $B$

Is there a common notation for the set of all injections from $$A$$ into $$B$$?

Some set-theorists use $$B^{(A)}$$, e.g., A. Levy in his book Basic Set Theory.

But some combinatorists use $$B^{\underline{A}}$$ or $$(B)_A$$, e.g. JMoravitz's answer in this question.

Some other combinatorists also use $$\mathrm{Inj}(A,B)$$, e.g., M. Aigner in his book Combinatorial Theory. But I don't like a notation of this kind, since I want something similar to $$B^A$$ or $${}^AB$$ which is commonly used to denote the set of all maps from $$A$$ to $$B$$.

Any suggestions for a notation are welcome.

• I don't think this question is suitable here.. Oct 24, 2019 at 4:03
• @Praphulla I'm sorry for that. If not, I will delete it tomorrow. Oct 24, 2019 at 5:07
• Personally I think terminology and notation are worth topics. I would be curious to read the answer to this question Oct 24, 2019 at 6:53
• About $B^{(A)}$: at least in group theory, it is frequently used in another meaning. Namely $A$ is a set, $(B,o)$ is a pointed set (typically a group), and $B^{(A)}$ is the set of finitely supported functions $f:A\to B$, that is, such that $f(a)=o$ for all but finitely many $a\in A$.
– YCor
Oct 24, 2019 at 11:29
• @YCor A. Levy also has a similar concept in his book, but he uses $\mathrm{exp}(B,A)$. I also think $B^{(A)}$ is not good for denoting the set of injections. Besides the reason you just mentioned, some combinatorists use $B^{(A)}$ for another meaning, namely rising factorials. Oct 24, 2019 at 11:54

The notation suggested by cardinal equalities such as

$$\begin{array}{l|l|l} \text{concept} & \text{notation} & \text{cardinality} \\ \hline \text{disjoint union of A and B} & A + B & |A + B| = |A| + |B| \\ \text{Cartesian product of A and B} & A \times B & |A \times B| = |A| \times |B| \\ \text{set of functions from A to B, also A \rightarrow B} & B^A & |B^A| = |B|^{|A|} \\ \text{set of permutations of A, also \text{Sym}(A)} & A! & |A!| = |A|! \\ \text{set of k-element subsets of A} & \binom{A}{k} & \left|\binom{A}{k}\right| = \binom{|A|}{k} \\ \text{set of k-element partitions of A} & \left\{{A \atop k}\right\} & \left| \left\{{A \atop k}\right\} \right| = \left\{{|A| \atop k}\right\} \end{array}$$

is

$$\begin{array}{l|l|l} \text{concept} & \text{notation} & \text{cardinality} \\ \hline \text{set of injections from A to B} & B^{\underline{A}} & |B^{\underline{A}}| = |B|^{\underline{|A|}} \end{array}$$

because the falling factorial

\begin{align*} |B|^\underline{|A|} = \frac{|B|!}{(|B| - |A|)!} \end{align*}

is precisely the number of injections from $$A$$ to $$B$$.

• Thank you. This is one of the notations mentioned in the post. Oct 26, 2019 at 2:06

If I take your question literally, it seems to me that the correct answer is simply "No". But I quite happily use $$B^A_{\neq}$$. Analogously, if $$A$$ and $$B$$ happen to be ordered, I write $$B^A_{<}$$ for the set of all strictly increasing functions from $$A$$ to $$B$$. For me, this works well.

• I didn't immediately notice this notation is not just arbitrary but views injections as maps between preserving the binary relation $\neq$.
– YCor
Oct 25, 2019 at 19:11
• With @YCor's convention, $B^A_<$ and $B^A_>$, the functions preserving $<$ and $>$, are the same. Is that really what's intended? May 19, 2021 at 20:41
• @LSpice I think so. It might look a bit weird maybe because it uses the same notation for the binary relations at the source and target. So it might be denoted $B^A_{\neq,\neq}$, and $B^A_{<_A,<_B}=B^A_{>_A,>_B}$ would both be the set of strictly increasing maps $(A,<_A)\to (B,<_B)$. The decreasing ones would be $B^A_{<_A,>_B}=B^A_{>_A,<_B}$. More generally, if $R-$ denotes the flip of a binary relation $R$, the when $R,S$ are binary relations on $A,B$, we get $B^A_{R,S}=B^A_{R-,S-}$.
– YCor
May 19, 2021 at 22:35