Strict weak ordering facts for kids

A strict weak ordering is a special way to put things in order. Imagine you have a group of items, like toys or books. A strict weak ordering helps you arrange them based on a certain rule.

For example, think about the rule "costs less than." If you have a toy car and a toy plane, the car might cost less than the plane. This rule helps you put them in order.

But what if you have two toy cars that are exactly the same price? The rule "costs less than" doesn't help you decide which one comes first. They are "equal" in terms of price. This is what a strict weak ordering handles: it orders things, but it also allows some items to be considered "equal" or "tied" based on the rule.

It's like when you sort words by how many letters they have. The word "cat" (3 letters) will always come before "tiger" (5 letters). But if you have "dog" (3 letters) and "fox" (3 letters), they both have the same number of letters. A strict weak ordering would put all 3-letter words before 4-letter words, but it might not care about the order of "dog" and "fox" among themselves.

Time also works this way. If two events happen at the exact same moment, they are "equal" in terms of time. But an event that happened yesterday will always come before an event happening today.

Understanding How Things Are Ordered

When we talk about ordering things, we often use a rule called a binary relation. This just means we compare two items at a time. For example, "is taller than" or "is heavier than."

What Makes It "Strict"?

The "strict" part means that if item A is related to item B (like A "costs less than" B), then B cannot be related to A (B cannot "cost less than" A). Also, an item cannot be related to itself (a car cannot "cost less than" itself).

What Makes It "Weak"?

The "weak" part is where it gets interesting for kids! It means that some items can be "tied" or "equivalent" according to the rule. Even if two items are different, they might be considered the same in terms of the ordering rule.

For example:

• Two different types of candy might cost the same amount.
• Two different students might have the same score on a test.
• Two different songs might be the same length in minutes.

In these cases, the strict weak ordering puts all the cheaper candies before the more expensive ones, or all the higher scores before the lower ones. But for items that are "tied," their exact order among themselves doesn't matter for the overall sorting.

Examples in Everyday Life

You see strict weak ordering all the time without even knowing the fancy name!

Sorting by Size

Imagine you have a pile of different-sized T-shirts. You want to sort them from smallest to largest.

• A small T-shirt is definitely smaller than a large T-shirt.
• But you might have two small T-shirts that are exactly the same size. They are "equivalent" in terms of size. The ordering puts all small T-shirts before medium ones, but the order of the two identical small T-shirts doesn't change the overall sorting.

Organizing Books by Author

If you organize your books by author, all books by J.K. Rowling would come before books by Rick Riordan (if you sort alphabetically by last name). But if J.K. Rowling wrote several books, the order of those books among themselves might not be strictly defined by the author rule. They are "equivalent" in terms of being by the same author.

Time and Events

When we talk about Time, events can happen at different moments.

• Your birthday last year happened before your birthday this year.
• But sometimes, two different things can happen at the exact same moment, like two friends saying the same thing at the same time. They are "simultaneous" or "at the same time." A strict weak ordering of events would place all earlier events before later ones, but events happening at the exact same time are treated as equivalent.

Strict weak ordering is a useful idea in computer science and mathematics because it helps us understand how to sort and organize information, even when some items are "tied" by the rules we set.