Saturday, 6 October 2012

Sets: An Introduction to Sets

A set is a very simple idea on the face of it, it is merely a collection or group; that is all. Sets are not objects in the real world; they are created within our minds not by our hands. An example of a set is all even numbers or all of the children in Europe or even the set of green bumblebees. But that is obviously not the sort of set we are interested in, we are only interested in ones that have interesting mathematical properties.

Sets by themselves may seem pretty uninteresting but when used in other areas of mathematics they show how powerful they truly are. They are
 used as a foundation from which the majority of mathematics can be derived, that is why they're so fundamentally important to mathematicians! Sets have to be defined in such a way that creates some list of numbers.

By convention, sets are denoted by capital letters, elements (read more in the next paragraph) are labelled by lower case letters. Beyond this the notation for sets is basic:
 you list each element, separated by a comma, and surround this list by curly braces, {}. For example, some set, Q, contains elements {p, q, r}.

There are a range of ways in which a set may be defined, either the items are defined using a semantic description for example: all of the odd integers or all prime numbers divisible by 4. Or you can define a set by listing the elements individually, for example: {2, 4, 8, 16}, with this version an ellipsis may be used to show that the set continues indefinitely in the same manner {3, 9, 27, 81, ...} like so.

A set constitutes of elements, an element is one distinct item that makes up a set. For example 1 is an element of the set of integers, the notation for an element, x, belonging to a set R is: x ∈ R. x  R means x is not an element of R.

A subset is a set that entirely lies within another set, for example all prime numbers are integers. 
A ⊆ B means every element of A is also an element of B and this is read as "A is a subset of B". An interesting point from this definition is that for any set, S, every element of S is clearly also in S thus S is a subset of S (S
 ⊆ S). This is a bit strange so we introduce something more thorough than subsets and that is proper subsets: A is a proper subset of B if every element in A is also in B and there is at least one element in B that is not in A; this has the slightly different notation of A  B.

The next two you may be more familiar with if you have done maths past GCSE level you may have encountered them, they are union and intersection. A union B is essentially A and B, the notation for this is 
A ∪ B, it is everything that is in A and B. A intersection B is where A and B overlap and the notation for this is A ∩ B.

The union of all of the natural numbers {1, 2, 3, ...} and all of the integers {-2, -1, 0, 1, 2, ...} will result in just the set of integers, the reason for this is that it does not matter if a set contains duplicate identical elements it is the same as long as the same elements are present, not how many. {1, 2, 3} is the same set as {1, 1, 2, 3, 2, 3} as the same elements are present in both. It then follows that 
A ∪ B = A + B - A ∩ B, which is a vitally important fact for combining sets.

You may also be interested in the size of infinite sets, which investigates that there are some infinities larger than others!

1 comment:

  1. Can you see that m divides a and b? And if that is the case, a and b actually have a difference of a multiple of m. So this means that, 49 ≡ 37 ≡ 25 ≡ 13 ≡ 1 (mod 12). You just add 12 to the number, you get another number which is congruent modulo 12. help me with math