Set Notation


In mathematics, a set is a collection of objects (in our case numbers). We can write a set using curly brackets, for example \(\{1, 5, 7\}\).

We say something is an 'element' of a set if it is contained in it, for example 1 is an element of the set \(\{1, 5, 7\}\). We write \(1\in\{1, 5, 7\}\) to mean this. Similarly, we can use the \(\notin\) symbol to mean that something is not an element of a set. The empty set is one with no elements, and it is denoted by the symbol \(\emptyset\).

If a set \(B\) is a subset of \(A\), then all of the elements of \(B\) are also elements of \(A\). We write \(B\subset A\) to mean this. The union of two sets is the set that contains all the elements of both sets, written as \(A \cup B\). The intersection of two sets is the set containing only the elements that are contained in both sets, written as \(A \cap B\).


Classification of numbers:

For A-Level, you should know basic sets of numbers in set notation:

 `bbb "N"` Natural numbers - the set of all positive integers: \(\{1,2,3,4,5,\ldots \}\) (different definitions may include \(0\), there is no definite answer as to whether it should or shouldn't be included).
`bbb "Z"`  Integers - the set \(\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}\)
`bbb "Q"`  Rational numbers - those which can be expressed as the quotient of two integers, i.e. in the form \(\frac{a}{b}\) where `a, b in bbb"Z"` .
`bbb "I"`  Irrational numbers - numbers that are not rational.
`bbb "R"` Real numbers - numbers that can represent a distance along a line.


Where `bbb "N" sub bbb "Z" sub bbb "Q" sub bbb "R"`


Expressing ranges in set-builder notation:

You may be asked to express a range in set notation. For example, if we had the range \(0<x<1\) for `x in bbb "R"`, we would write this as:

`{x in bbb "R" | 0<x<1}`

Where the `x in bbb "R"` indicated that we are working in real numbers, and the vertical bar "\(|\)" (sometimes "\(:\)" is used instead) means "such that". I.e. the set notation says "x is an element of reals such that \(0<x<1\)".

If you wanted to write a more complicated range, such as \(0<x<1\) or \(x>2\), you could write:

`{x in bbb "R"| 0<x<1}uu{x in bbb "R"| x>2}`

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