Set Language and Notation

Set Language and Notation is a way to describe and communicate about collections, or ‘sets’, of numbers or items. This language is used in mathematics to clearly define and discuss these collections.

A ‘Set’ is a collection of unique elements or members. These members can be anything  numbers, letters, shapes, etc.

Notation for a set usually involves curly brackets  { }. For example, a set of the first three positive integers would be written as {1, 2, 3}.

The ‘Empty Set’ or ‘Null Set’ is a set that contains no elements. It is denoted as {} or Ø.

If an element is part of a set, we say it ‘belongs to’ the set. This is signified with the symbol ∈. For instance, if A = {2, 4, 6, 8}, we can say 4 ∈ A.

The ‘Universal Set’ is the set of all possible elements in a particular context. It is often denoted by the symbol 𝒰.

A ‘Subset’ is a set where all its elements also belong to another set (the ‘parent set’). For example, if B = {2, 4}, B is a subset of A in the previous example.

If a set includes all the elements of another set, it is a ‘Superset’. Using the previous example, A is a superset of B.

Notation for subset and superset are ⊆ and ⊇ respectively.

The ‘Intersection’ of two sets is a set of elements that both original sets have in common. It’s shown with the symbol ∩. If C = {4, 6, 8, 10}, then A ∩ C = {4, 6, 8}.

The ‘Union’ of two sets is a set that includes all the elements from both original sets, with no repetitions. It’s shown with the symbol ∪. Following the previous example, A ∪ C = {2, 4, 6, 8, 10}.

The ‘Difference’ of two sets (set A  set B) results in a set containing elements that are in A but not in B. Using the sets A and B defined earlier, A  B = {6, 8}.

A ‘Complement’ of a set, notated as A’, includes all the elements in the Universal Set that are not in the defined set.

‘Cardinality’ refers to the number of elements in a set. The cardinality of set A defined above would be 4 since it includes 4 numbers.
Remember: Practicing notation and understanding set language greatly aids mathematical understanding and problem solving. It’s especially helpful with probability, number theory, and some parts of algebra. Reviewing sets often and applying them in different contexts will solidify understanding.