The definition of “into” will call back to my A to Z piece on “bijections”. It particularly call on what mathematicians mean by a function. When a mathematician talks about a functions she means a combination three components. The first is a set called the domain. The second is a set called the range. The last is a rule that matches up things in the domain to things in the range.
We said the function was “onto” if absolutely everything which was in the range got used. That is, if everything in the range has at least one thing in the domain that the rule matches to it. The function that has domain of -3 to 3, and range of -27 to 27, and the rule that matches a number x in the domain to the number x3 in the range is “onto”.
Not every function is onto, though. The range can have elements that aren’t matched by anything in the domain. Imagine the function had a domain of -3 to 3, and a range of -50 to 50, with the same rule matching the number x in the domain to the number x3 in the range. Then we’d have an “into” function. If there’s even a single thing in the range that isn’t matched to something in the domain, then the function is into the range.
One might ask why have a range that contains stuff that the function doesn’t need. It’s easy to suppose that laziness is part of it. If a function has a particularly complicated rule it might be too much bother to work out that it really only needs the numbers between -22 + π and the square root of seven, excepting the numbers between -2 and 1, and for some reason also excepting the square root of five, or anything that has a “2” in the fifth through eighth places after the decimal point. Easier to just say the range is -22 to 7, or maybe all the real numbers, and not worry that we list too much. We can’t exhaust the numbers. We might exhaust our audience’s patience.
But having more stuff in the range than we need can help us understand a problem better. When we model a physical problem, for example, it’s often useful to start with a volume of space that represents all the possible values the variables could have. Those variables are usually things like positions and speeds. For example, the system could be a car in traffic. The variables would be the mile marker the car’s at and the speed at which it’s travelling. And the differential equation can be rewritten as a function. Its domain is the possible values the variable could have. The range is the same thing. The rulesays that if the car was here when we started, then after (say) a minute it will be at this new position and have this new speed.
In this case, it makes sense for the range to be the same set as the domain. The domain was the set of all the possible values that might describe a car in traffic, and we’re still describing a car in traffic. But the function might be “into”. There could be sets of values that we might start from, but can’t get back to. Imagine that there’s a traffic accident that closes off the entrance to the road at exit 1. As time goes on, there’ll be an ever-growing swath of road that has no cars on it. There are positions that cars can’t access anymore. The function describing how traffic changes over the course of a minute has become an “into” function.
We might want to talk about the part of the range that is needed . That would be part of the range that the rule matches to something in the domain . We call that region the “image” or “projection” of the domain under the function. A function is always “onto” the image of its domain. That’s what the image of the domain means.