Kitchen jars, Cinderella and functions of a variable

Once upon a time in a very distant kitchen, a cupboard full of pots and lids.
Whenever someone needed to keep something, a journey begins in haste behind compatible pots and lids in that collection.
But there were lids and pots of all shapes and sizes, and no pot or lid was repeated.
The search for pot and lid ended when they found a lid close to the standards expected to close the pot.
If the lid was just to cover, then it could be bigger than the pot.
If it was to close it, a slightly smaller lid with “that tight one” would do (or sometimes the lid would crack when closing).
If a pot or lid broke, it would go to waste but your partner would stay in the closet.
If the situation involved moving the food (for example, taking the lunch box in the backpack), the search was even harder, we needed one that would close perfectly.
However, even when lids and pots met and lived happy moments together, after the dishes were washed, they all returned to the same closet of chaos, where the new journey would begin.

Do you notice any similarity between this drama and the story of Cinderella?

In this drama, from time to time, for the joy of the little pots, a fairy godmother appeared, full of her “Bibidi Bobidi Bu!” he gathered the pots with their lids, bringing the pairs together and giving them moments of happiness. But the magic ends at midnight, when after cleaning they return to the closet and lose hope of forming to meet their peers again.

But what the hell does that have to do with math? Did this period of quarantine make me lose the blog line and now I am anthropomorphizing cooking pots? Maybe so, but today’s post has to do with exactly this theme and functions of a variable. Yes, the inspiration came while organizing the pots at my mother-in-law’s house (for you to see how science can participate in our daily lives).

The problem of organizing pots with lids involves in particular the formation of bijector relations between the set of pots and the set of lids. For example, I love to have many equal pots, I have about 20 pots of just 3 types, so my set of pots is can be described as: {A, A, A, A, A, A, A, A, B, B , B, B, B, B, B, B, B, B, C, C} and my set of covers {Z, Z, Z, Z, Z, Z, Z, Z, Y, Y, Y, Y , Y, Y, Y, Y, Y, Y, X, X}.

When you find a Z-type cover, you just need to check its compatibility with:
Type A pot – Type Z lid;
Type B pot – Type Z lid;
Type C pot – Type Z lid.

As a rule, we assume that each type of lid fits perfectly with only one type of pot. So, if the same lid can be used for two pots, then they are the same type.

However, in a less organized scenario, we may have a situation similar to the tale at the beginning of this post, in which in a kitchen we have all kinds of different pots and lids. Just to exemplify, we will deal with a case with 13 types of pots and their respective 13 types of lids, all types with only one unit. Let’s call the pots in the set {A, B, C, D, E, F, G, H, I, J, K, L, M} and the lids {Z, Y, X, W, V, U, T, S, R, Q, P, O, N}.
Just to facilitate future notations, let’s say that the relationship of the pots and lids is this:
A-Z B-Y C-X D-W E-V F-U G-T H-S I-R J-Q K-P L-O M-N

When finding a type P cover we should check its compatibility with:
Type A pot – Type P lid;
Type B pot – Type P lid;
Type C pot – Type P lid;
Type D pot – Type P lid;
Pot type E – Lid type P;
Type F pot – Type P lid;
Type G pot – Type P lid;
Type H pot – Type P lid;
Type I pot – Type P lid;
Type J pot – Type P lid;
Type K pot – Type P lid.

We interrupted this search in pot K, since we found a pot with perfect fit (pot K and lid P)

But if our conditions allow a pot with close compatibility, we can for example close it after testing the type P lid with the type H pot, accepting that this one, for example with a little smaller than the pot, serving since we apply it a little of force to fit it.s just by giving that tight.

However, now we have a greater contingency, because the type H pot is not perfectly compatible with that lid, that is, S (the real lid of pot H), it will be incompatible to fit in pot K (the real pot of lid P).

So even if the search for the pot of lid S occurs with all the remaining 12 pots, none of them will have the perfect compatibility, requiring the subject to decide again between a slightly larger or smaller pot.

If we choose pot K, we close this cycle with two pots improperly capped (K-S; H-P).

But if we choose a pot other than K (for example, pot G), then there is now the lid T left (true combination of pot G), taking the combinations G-S and H-P.
In this way, we have pot K and cap T with no matching pair.

Note that the wrong choices can close a cycle or extend it, in the second case there is the possibility of compatibility errors increasing even more. This is because there is no longer any guarantee that the little more or less used in one combination will be compensated inversely proportional in the other.

This situation is the result of a rupture in the initial two-way relationship between pots and lids, that is, each pot has its respective lid. When we allow too many caps for a pot and vice versa, we have a relationship that cannot be described as a function of a single variable, for example:
pot (1) leads to the lid (1);
pot (2) leads to the lid (2);

But from this error, we arrived that the pot (1) leads to the lid (1) and the lid (2).

Thus, as in the situation presented, if the function does not have a relationship between all the elements of each set or this relationship is not unique, as in the case of lids and pots, it cannot be expressed as a function of a variable.

To understand why functions of a variable need to have a bijector relationship between sets, imagine the function of converting centimeters to inches. Whatever the measurement in centimeters, we must convert to a single measurement in inches, and any measurement in inches must have a corresponding measurement in centimeters.

If, for example, we have two different inch values ​​for the same measurement in centimeters, then conversion is not a function of a variable (did it become clearer where this is the problem of an entry leading to two different responses?).

Back to the problem of lids and pots, how to solve this? Index lids and pots:
pot (1) – lid (1);
pot (2) – lid (2);
pot (3) – lid (3)

pot (n) – lid (n)

What if you have several identical pots and lids?

Easy, number all the same with the same number, because the numbering only concerns the compatibility of the pots and lids, not necessarily how many we have of each type.

For example, the screen of two 14-inch computers will have the same conversion to centimeters, even though their make, model and other aspects are different. This in no way affects the relation being bijector (in mathematics, repeated elements belonging to a set are generally treated as the same element).

To facilitate the management of our pots-lids function, keep the lids and pots (if possible) in an orderly way, so when we find pot 15, we won’t need to search among all the lids which one has the number 15, we can from any cover found, find out if number 15 is in front of or behind it. In the same context of unit conversion tables, we expect the 14 inch unit to be listed after the 13 inch unit and before the 15 inch unit (have you ever wondered if they were scrambled, the work to find a measurement?)

Unique pots and lids can be left without indexing (but indexing won’t hurt either).

But what to do with unique lids and pots? (In my case, I had a lot of lids left, but no pots)

Keep them in a well-hidden place, so that no one can innocently mix with peers and ruin our beloved bijector relationship that admits that we can transform pot-lids into a function of a variable.