Set
From the Simple English Wikipedia, the free encyclopedia that anyone can change
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).
In the following sections, a bag is a shopping bag.
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[change] What to do with sets
[change] How to tell others about the set
Usually, when things are put 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)
[change] Element of
Various things can be put into a bag. Later on, a valid question would be if a certain thing is in the bag. Mathematicians call this element of. Something is an element of a set, if that thing can be found in the respective bag.
[change] 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.
[change] 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.
[change] 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.
A set can have an infinite number of elements. One such set is the set of natural numbers. Some sets with an infinite number of elements are bigger (have a bigger cardinality) than others. There are more real numbers than there are natural numbers, for example.
[change] 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 as a smaller bag, inside the bigger bag. In the 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.
[change] Combining sets
There are different ways to combine sets.
[change] Unions
The Union of two sets is a set that contains all the elements of both sets. It is like taking several shopping bags, and putting all things from them into a bigger bag.
[change] 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.
[change] 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. Taking the example from above, this would be potatoes and sausage.
[change] Special sets
Some sets are very important to mathematics. They are used very often. One of these is the empty set. Many of these sets are written using blackboard bold typeface, as shown below. Special sets include:
, denoting the set of all primes.
, denoting the set of all natural numbers. That is to say, = {1, 2, 3, ...}, or sometimes = {0, 1, 2, 3, ...}.
, denoting the set of all integers (whether positive, negative or zero). So = {..., -2, -1, 0, 1, 2, ...}.
, denoting the set of all rational numbers (that is, the set of all proper and improper fractions). So, 
. For example, 
and 
. All integers are in this set since every integer a can be expressed as the fraction 
.
, denoting the set of all real numbers. This set includes all rational numbers, together with all irrational numbers (that is, numbers which cannot be rewritten as fractions, such as π, e, and √2).
, denoting the set of all complex numbers.
Each of these sets of numbers has an infinite number of elements, and 
. The primes are used less frequently than the others outside of number theory and related fields.
[change] 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:
[change] 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.
[change] Further reading
The following are books about sets. They may not be easy to read though:
- Halmos, Paul R., Naive Set Theory, Princeton, N.J.: Van Nostrand (1960) ISBN 0-387-90092-6
- Stoll, Robert R., Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4
- Allenby, R.B.J.T, Rings, Fields and Groups, Leeds, England: Butterworth Heinemann (1991) ISBN 0-340-54440-6
