What's the precise meaning of the expression "induced by" in mathematics? [duplicate]

It's been more than once I've found this expression "induced by", in a sentence of the form "$X$ is induced by $Y$, in mathematics and computer science. I usually associate "induced by" with "generated by". However, I am not always confident regarding its meaning.

For example, in the following sentence

If a planar subdivision is induced by $n$ line segments...

What's the precise meaning of "induced by", in general, and in the sentence above?

• I think its precise meaning will vary from context to context, but I think "generated by" is a good synonym. "Produced by" also works in some contexts. Commented Mar 23, 2018 at 15:06
• If well used, it means that the author/speaker has in mind a well defined general procedure by the means of which, if given Y, you can produce X. This sounds very hazy, but i guess that is precisely why the expression is so handy. As @angryavian says, in a given context it should be clear. Commented Mar 23, 2018 at 15:09

First, "induce" is a perfectly cromulent English word. The second definition that Google gives is relevant here:

bring about or give rise to.

In basic vernacular English, it is reasonable to say that "$A$ induces $B$" when $A$ causes $B$, though I think that there is a connotation of indirectness (i.e. there might not be that $A$ directly causes $B$, but $A$ creates the conditions for $B$). In mathematics, this is the definition that is generally meant. When we say that "$A$ induces $B$," we typically mean that $A$ gives rise to $B$, typically in some canonical manner.

For example (in an area with which I am more familiar), we often say that a "metric induces a topology". What this means is the following: if $(X,d)$ is a metric space, then the open balls, i.e. the collection $$\mathscr{B} := \{ B(x,r) : x\in X, r> 0 \},$$ where $B(x,r) := \{ y \in X : d(x,y) < r \}$, forms a basis for a topology on $X$. The topology generated by this basis is the topology induced by the metric. That is, the metric gives rise to this topology.

After a bit of Googling, a "planar subdivision induced by a set of $n$ line segments" seems to make sense in a similar way. Near as I can tell, a planar subdivision is a partition of the plane, i.e. a division of the plane into a collection of mutually disjoint sets whose union is the plane. A partition has more structure than just a collection of line segments, but a collection of line segments can give rise to a partition in a canonical manner. It is therefore appropriate to say that such a partition is induced by a collection of line segments.

• +1, basically the answer I would have written. Give rise to is a very reasonable synonym for induce; indeed, I think I see that used in math writing a lot too. A link that explains your “cromulent” reference would improve the answer even more. Commented Mar 23, 2018 at 16:27
• I've added a reference, but I prefer to leave the cultural significance as a reward for those that seek out further information. ;) Commented Mar 23, 2018 at 16:42
• Its significance embiggens us all. Commented Mar 24, 2018 at 1:47

In the specific case, the $n$ line segments can be extended uniquely to lines, that give a planar subdivision (Edit: see comments below). In general, as mentioned in the comments, there is no literal interpretation that always works. Sometimes we have a "smaller" thing that can be extended uniquely (as in this case), sometimes we have a " bigger" thing that can be reduced uniquely (as for the induced or relative topology on a subset of a topological space), and sometimes neither of the two. A good use of the word requires that it is clear from the context what is meant. The idea is always to adapt /modify something given in a canonical / unique way to get what we need.

• As far as I can tell, a planar subdivision is a partition of the plane subordinate to a planar graph. Why are you extending the segments to lines? See, for example, this or this. Otherwise, your answer seems spot on. Commented Mar 23, 2018 at 15:23
• @XanderHenderson My bad, I just assumed, for some reason, that a planar subdivision was a subdivision of the plane by lines. In any case, I imagine that to get a planar graph one extends the segments and takes the intersection points of the induced lines as vertices of a graph, and the segments between two vertices as edges. Or does it mean something different? Commented Mar 23, 2018 at 15:34
• I'm not familiar with the terminology, but google seems to indicate that a planar subdivision is a partition of the plane subordinate to a graph. The example I linked above uses a map as an example---every point on the plane lives in some country or another, thus the boundaries of the countries (the graph) induce a planar subdivision (each point belongs to some country, countries are some kind of equivalence class). I'm a bit confused about what we do with the boundaries themselves, but this is far from my area of expertise. Commented Mar 23, 2018 at 15:41
• @XanderHenderson So it is for me. Thank you in any case for the comment Commented Mar 23, 2018 at 15:44