Set
From Simple English Wikipedia, the free encyclopedia
A Set is a concept from mathematics. A set is like a bag, that can hold things. A set cannot hold a certain item more than once. Either that item is in the set or it is not. Structures from mathematics, that are like sets in quite a few ways, but can hold a certain type of item more than once are called multisets (or indeed, bags).
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[edit] What to do with sets
[edit] How to tell others about the set
Usually, when you put things into a bag, all the things that are put in have something in common. If someone else needs to get the same set, there are different options on how to tell them:
- All elements could simply be stated (like a shopping list).
- Some common thing could be stated (eg. green vegetables)
[edit] Element of
You can put different things into the bag. Later on, you ask whether a certain thing is in the bag. Mathematicians call this element of. Something is an element of a set, if you find that thing in the respective bag.
[edit] Empty set
Like a bag, a set can also be empty. The empty set is like an empty bag: it has no things in it.
[edit] Comparing sets
Two sets can be compared. This is done by looking at two different bags. If they contain the same things, they are equal.
[edit] Cardinality of a set
When mathematicians talk about a set, they sometimes want to know how big a set is. They do this by counting how many elements are in the set (how many items are in the bag). The cardinality is a simple number. The empty set has a cardinality of 0, since there are no things in the respective bag.
[edit] Subsets
A set can have a large number of elements. Like a pretty full, large bag. Some of these elements perhaps have some other things in common, other than that they are all in the bag. Mathematicians call this a subset. It can be thought of like a smaller bag, inside the bigger bag. In your shopping bag, there might be a bag of vegetables and a bag containing meat. Those two sets would then be subsets of the bigger set.
[edit] Combining sets
There are different ways to combine sets.
[edit] Unions
The Union of two sets is a set that contains all the elements of both sets. Its like taking several shopping bags, and putting all things into a bigger bag.
[edit] Intersections
The intersections of two sets is a set that contains all the elements, that are in both sets. If two people went shopping independently, the intersection is all the things that both of them bought: if one bought apples, carrots, and potatoes and the other bought apples, carrots and sausage, the intersection would be apples and carrots.
[edit] Complements
The complement is like the difference of two sets. It's like saying I want all things that are in one bag, but not in the other bag.
[edit] Paradoxes about sets
A mathematician called Bertrand Russell found that there are problems with this theory of sets. He stated this in a paradox called Russell's paradox. An easier to understand version, closer to real life, is called the Barber paradox:
[edit] The barber paradox
There is a small town somewhere. In that town, there is a barber. All the men in the town do not like beards, so they either shave themselves, or they go to the barber shop to be shaved by the barber.
We can therefore make a statement about the barber himself: The barber shaves all men that do not shave themselves. He only shaves those men (since the others shave themselves and do not need a barber to give them a shave).
This of course raises the question: What does the barber do each morning to look clean-shaven? This is the paradox.
- If the barber does not shave himself, he will follow the rule and shave himself (go to the barber shop to have a shave)
- If the barber does indeed shave himself, he will not shave himself, according to the rule given above.
