Bijective Function: Intro (Math 0)

Com­par­ing Amounts

Con­sider the Count von Count. He cares only about count­ing things. He doesn’t care what they are, just how many there are. He de­cides that he wants to col­lect items into plas­tic crates, and he con­sid­ers two crates equal if both con­tain the same num­ber of items.

Equivalent Crates

Now Elmo comes to visit, and he wants to im­press the Count, but Elmo is not great at count­ing. Without count­ing them ex­plic­itly, how can Elmo tell if two crates con­tain the same num­ber of items?

Well, he can take one item out of each crate and put the pair to one side.

Pairing one pair of items

He con­tinues pairing items up in this way and when one crate runs out he checks if there are any left over in the other crate. If there aren’t any left over, then he knows there were the same num­ber of items in both crates.

Pairing all items in one crate with items in the other

Since the Count von Count only cares about count­ing things, the two crates are ba­si­cally equiv­a­lent, and might as well be the same crate to him. When­ever two ob­jects are the same from a cer­tain per­spec­tive, we say that they are iso­mor­phic.

In this ex­am­ple, the way in which the crates were the same is that each item in one crate could be paired with an item in the other.

Pairing all items in one box with items in the other

Bijection between crates

This wouldn’t have been pos­si­ble if the crates had differ­ent num­bers of items in them.

Different numbers of items

No way to pair items

When­ever you can match each item in one col­lec­tion with ex­actly one item in an­other col­lec­tion, we say that the col­lec­tions are bi­jec­tive and the way you paired them is a bi­jec­tion. A bi­jec­tion is a spe­cific kind of iso­mor­phism.

Bijection between crates

Note that there might be many differ­ent bi­jec­tions be­tween two bi­jec­tive things.

Another bijection between crates

In fact, all that count­ing in­volves is pairing up the things you want to count, ei­ther with your fingers, or with the con­cepts of ‘num­bers’ in your head. If there are as many ob­jects in one crate as there are num­bers from one to seven, and there are as many ob­jects in an­other crate as num­bers from one to seven, then both crates con­tain the same num­ber of ob­jects.