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Element of a Set:  An object or idea in a set is called an element (or member) of the set.

 

Notation:  The symbol ∈ is used to denote that an element is a member of a set and ∉ is used to denote that an object is not a member of a set.

 Example:  For set A = {1, 2, 3}, 1 ∈ A, but 12 ∉ A.

 Example: For S = {x : x is a state bordering Minnesota}, Iowa ∈ S, but Alabama ∉ S.

 Empty Set:   The empty set (or null set) is a set that has no members.  

Notation: The symbol Ø  is used to represent the empty set, { }. 

Example:
        Ø  = The collection of people attending MSUM who are 200 years old (verbal)    
        Ø = { }   (roster)
        Ø = {x : x is a person attending MSUM who is 200 years old.}  (set-builder)

Note:  {Ø} does not symbolize the empty set; it represents a set that contains an empty set as an element and hence has a cardinality of one.

 Equal Sets. Two sets are equal, if they have exactly the same elements.

Example:  {a, c, t} = {c, a, t} = {t, a, c}, but {a, c, t} ≠ {a, c, t, o, r}.

 Example:  {x : x is a letter in the word “book”} = {b, o, k}, but {b, o, k} ≠ {b, o, t}.

Note:  The order of the elements inside the brackets in the roster form does not matter.
When listing elements in the roster form, we do not repeat elements inside the set brackets.