# When are non-homogeneous sets first encountered by students?

My 5-year-old son recently brought home (from kindergarten) a worksheet that he had done at school. 6 boxes were given. In one box were 3 kites, in another, 2 bears, in another 8 drums, and so on. Each box had a list of 3 numbers at the bottom to choose from. The instructions were:

“Count the objects in each set. Draw a circle around the right number.”

As we all know, such exercises are common for young students, but I have always wondered whether such exercises were teaching something that would later have to be unlearned, namely, the false idea that members of a set are always all instances of the same thing. Anyway, at the very least we should ask when in their academic career students finally encounter the truth that the members of a set can be things of different type. 8th grade? senior year of high school? College? Only math majors in college after their first year?

• Depending on the materials/instructor, it might well come this year. You could have students count people in their family (all "people" but not all the same in the way that I imagine, e.g., the depicted kites are). Or you might have students look at a box with cats and dogs, and have them count the cats, then count the dogs, then count the total number of animals. See, for example, here on p. 13 (under Pre-K). – Benjamin Dickman Dec 13 '14 at 2:44

It is very early primary school that they learn that sets can hold instances of different things. If they are of the same thing, then they are not really sets - or rather they have cardinality 1 as sets can't have repetition.

However, It is a good that they learn early on that set should contain things of the same type. A set of 3 apples can be unioned (added) with a set of 2 oranges, but what would the result be? Not a set of apples or oranges, but a set of 5 pieces of fruit. And then add in a dog, and you have a set of 6 formerly alive objects. Add in a cloud...

The more varied the type of thing in the set, the less you can do with it. For instance, you could have a set of a person, a name, a number, and mathematical equation and the set of prime numbers, but I can't imagine what you would ever do with it. What would be the point of counting it (5 items)? What kind of set would you intersect it with?

In Computer Science, arrays, lists, or sets are often typed to prevent randomly heterogenous sets. Tuples, records, or structs may contain heterogenous objects, but they can not be manipulated as sets.

• The importance given to heterogeneous sets seems to me to be an inadequate consequence of the choice of set theory as a foundation of mathematics. Maybe a type-oriented approach as homotopy type theory might change that. – Benoît Kloeckner Dec 14 '14 at 19:20
• (Why is the dog "formerly alive"?!) – Benjamin Dickman Dec 14 '14 at 20:03
• @BenjaminDickman the picture of it on the worksheet did not look very animated – Richard Dec 14 '14 at 22:40
• @BenjaminDickman, are picked pieces of fruit still alive? (I don't know.) Maybe it was just the most general descriptor that came to mind for the elements of a set containing fruit and a dog! – LSpice Dec 27 '14 at 17:09
• “I can't imagine what you would ever do with it” - "A failure of imagination is not an insight into necessity." -- Patrick Barrow – Mike Jones Mar 4 '16 at 23:51