What Are Sets?

A set is a collection of things. Specifically, it is a collection of mathematical identities. For example, mathematical numbers such as 1,2,3. This is a set.

The mathematical symbol for set is {1,2,3}. Let us call this set, A. A now stands for {1,2,3}. The numbers 1,2 and 3 are elements of the set A. Elements are separated by commas. The mathematical symbol for an element is - without the dash at the beginning. Therefore, 1 is an element of A... 1 A. This is the same with the numbers 2 and 3. Sets have a cardinality. Cardinality is the size of a set, meaning the number of elements it contains. The cardinality for set A is 3 since it has 3 elements. |A| symbol is ued to represent cardinality.

Therefore, |A|=3. Note that the set

{ 3,1,2}=A as well. There are some important sets that have their own symbols. We have decided to name our set, set A. The set {1,2,3,...}=N

The elipsis repreents infinity. N is the set of all natural numbers. The set {...,-2,-1,0,1,2,...}=Z. Z represents integers. Q reprents rational numbers, numbers which can be expressed as fractions. The square root of 2 is an irrational number. You might consider the idea that it can be expressed as a fraction. However, for a number to be in the set Q, it has to be able to be expressed as a fraction where the denominator is a a non-zero integer. This leads to another standard notation, the colon.

{x N:x is even}

This would be read as x is an element of the set of natural numbers such that x is even. The colon is used to replace the words "such that.'

So after this introduction,one might consider, why do we need to know this. This might bring memories of that 1 student who would always ask this question. However, it is a valid thought. Mathematics is a language. It might seem like a striking statement, almost an oxymoron. Maths is associated with numbers. Language has connotations of words. So how do words and numbers come together in this sentence? Well, without criticising the schooling system, i would like to offer my explannations. Maths is an indeeed a very interesting subject. However, it`s image has been tarnished with difficulty, numbers and tiresome problems to solve. Mathematics is mucu more than that. What we are missing in the modern area, is a lack of curiousity and willingness to explain and discover observations ourselves. This maybe due to our reliance on other`s information. To sum it up, it is our loss of thirst for knowledge and understanding, it is lack of asking "why?"

Why do we use (x,y) for co ordinates. Why do we have to integrate? How was this method discovered? What is the proof for this method? Mathematical understandinf relies heavily on proofs. When studied at university level, the proofs topic will spark an interesting in mathematics. It comes to realisation that all the methods and theoreoms and theories originated from somewhere and we begin to ask questions such as why and how. There are some students who understand that maths and other subjects have interesting origins and are fascinating subjects. However, schooling systems do not allow for their curiosities to be fed. The schooling system is focused on exam after exam after exam. One ladder to another. "Do we need to know this for the exam?" This is a question that has become all too common.

Without going off topic, why do you need to know what you just have hopefully learnt. Sets are vital. They are part of the mathematical langauge. You need to understand the langauge of maths in order to understand mathematics itself. For understanding proofs and origins of theoreoms, sets are foundational.