A stick for every cow

The idea

Associating each object of one collection with an object of another, so that no two objects are mapped to the same object, is called a one-to-one mapping.

Did every cow come home?

Long ago there were no number names. A farmer keeps one stick for each cow in the herd: 6 cows, 6 sticks. In the evening, 5 cows come home.

How can the farmer find out, without any number names, whether a cow is missing?

• A stick will be left without a cow
• Every stick will have a cow
• It cannot be found out

Pair every cow with a stick

Tap a cow, then tap a stick, to pair them. Each stick can take only one cow.

One stick for each cow

Associating each object of one collection with an object of another, so that no two objects are mapped to the same object, is called a one-to-one mapping.

A stick is left over, so one cow has not come home. The sticks tell us the number of cows, with no number names.

The same trick compares two herds. Pair my cows with the neighbour's cows. The cows left without a partner show who has more, and by how many.

Notes

Associating each object of one collection with an object of another, so that no two objects are mapped to the same object, is called a one-to-one mapping.

Check yourself

Which one is a one-to-one mapping of 3 cows to 3 sticks?

The farmer has 6 sticks, one for each cow that left. 4 cows come home and are paired with sticks. How many cows have not come home?

Answer: 2

6 sticks, 4 paired. 2 sticks are left over, so 2 cows are missing.

I have 6 cows and my neighbour has 9 cows. We pair them one-to-one. How many of the neighbour's cows are left without a partner?

Answer: 3

6 of the neighbour's cows are paired with mine. The other 9 − 6 = 3 are left, so the neighbour has 3 more cows.

After pairing, every cow has a stick and 2 sticks are left over. What do we know?

• Cow 1 → stick A, cow 2 → stick A, cow 3 → stick B. Two cows are mapped to the same stick A. In a one-to-one mapping each stick takes only one cow.
• Cow 1 → stick A, cow 2 → stick B, cow 3 → stick C — correct. Yes! Every cow has its own stick, and no stick is used twice.
• Every cow → the longest stick. All the cows are mapped to the same stick. Each cow needs a different one.
• There are 2 more cows than sticks. All the cows were paired, so no cow is left over. The leftover items are sticks.
• There are 2 more sticks than cows — correct. Yes! The 2 leftover sticks are the extra ones.
• There are equally many cows and sticks. If they were equally many, nothing would be left over.