5Roster notation

The simplest way to write down a set is to write down a list of its elements. To denote that we’re writing a set, we enclose the elements in curly brackets, ‘{’ and ‘}’:

For example, the notation {7, 8, 9, 10} denotes the set which contains the numbers 7, 8, 9 and 10, and nothing else.

The order in which the elements are written down doesn’t matter. Thus, {7, 8, 9} = {8, 9, 7}. This is because, for a set, the relevant elementhood facts are just: is this object an element of the set or not? Membership doesn’t come in any particular order, and an object can’t be an element of a set twice. It just is, or isn’t, an element, and there’s nothing further to say.

By convention, if we use roster notation while writing a given element multiple times, we simply assume that the element is a member of the set and treat it as though we had only written it once. Thus {7, 8} = {7, 7, 7, 7, 8, 8, 7, 8, 8, 7, 8}.