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In this snippet from Knuth's Concrete Mathematics, I'm not sure how we would know the range of $j, k$: enter image description here

On the left hand side of the above equation, we just are given that $1 \leq j, k \leq 3$ then how do we know that $1 \leq j \leq 3, 1 \leq k \leq 3$?

I assumed that $j$ would increment infinitely from 1 and that $k$ doesn't have a place to begin since there's originally no lower bound...

Katie Melosto
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"$1 \leq j, k \leq 3$" is just questionable notation, it's supposed to be interpreted as "both $j$ and $k$ are between $1$ and $3$ inclusive", so yeah, it does mean $1 \leq j \leq 3 \ \wedge 1 \leq k \leq 3$.

I've seen this more commonly written as $$\sum_{j,k=1}^3 a_jb_k$$ though that's similarly questionable.

Magma
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  • It is not really questionable notation, as long as one is consistent about it. There are numerous examples of "$x,y ◁ z$" or "$x ◁ y,z$" where "$◁$" is a relation: "$x,y ∈ S$", "$p \mid m,n$", "$a,b ≠ 0$", ... Of course, ideally, notation should be defined before used. – user21820 May 17 '20 at 05:01
  • I know it's established, but that makes it only slightly less questionable. – Magma May 17 '20 at 21:59
  • It is not questionable if it is clearly defined and unambiguous. Concise notation is crucial in mathematics. If you think otherwise, please demonstrate your preferred way of writing "$∀x,y,z{∈}S \ ( \ x<y<z ⇒ x<z \ )$". – user21820 May 18 '20 at 04:05
  • It's clearly defined, (sort of) unambiguous, established, and I use it often, but that doesn't make it unquestionable yet. – Magma May 18 '20 at 11:48
  • Hahaha. Okay. =) – user21820 May 18 '20 at 15:09