Types of Sets
Universal sets:The set containing everything
currently under consideration. Symbolized with U
Empty sets or Null Set: The set with no elements.
Symbolized ∅, but {} also used.
Singleton Set: A set with exactly one element.
Example: {1}
A common error is to confuse the empty set ∅ with the set {∅}, which
is a singleton set. The single element of the set {∅} is the empty
set itself! A useful analogy for remembering this difference is to
think of folders in a computer file system. The empty set can be
thought of as an empty folder and the set consisting of just the
empty set can be thought of as a folder with exactly one folder
inside, namely, the empty folder
Equal and Equivalent Sets: Two sets are equal if
and only if they have the same elements.Therefore, if A and B are
sets, then A and B are equal if and only if ∀x(x ∈ A ↔ x ∈ B). We
write A = B if A and B are equal sets.
Example: The sets {1,3,5} and {3,5,1} are equal,
because they have the same elements.
Note It does not matter if an element of a set is
listed more than once; so {1,3,3,3,5,5,5,5} is the same as the set
{1, 3,5} because they have the same elements.
Membership of sets: As established above a set
contains element, thus, if an element "a" is found in the set A, we
say that "a" is a member of A which is denoted as a ∈ B, however if
"b" is not a member of B it is denoted as b ∉ B.
Note: Sets can have other sets as members.
Example: The set {N,Z,Q,R} is a set containing four
elements, each of which is a set. The four elements of this set are
N, the set of natural numbers; Z, the set of integers; Q, the set of
rational numbers; and R, the set of real numbers.