More about Sets
What is a set?
• A set is a well defined
collection of objects.
• Is S = {some numbers} a
set?
• Is W = {Fred, George, Ron}
a set?
What is in a set?
• The objects in a set are
called elements or members.
• We write 3S, or
6  {1, 2, 3, …}
• Also, Harry  W
How do you write a
set?
• They are always enclosed by
brackets and the elements
are separated by commas.
The series of dots (…) are
called ellipses and mean
“the pattern continues”.
How do you write a
set?
• Sets can be written by rule
S = {x| 0 < x < 10 and x  I}
• or by roster
S = {1, 2, 3, 4, 5, …, 9}
More details...
• Two sets are equal iff they
contain the same elements.
• If A = {2, 3, 4, … 11} and if
B = {x|1 < x < 12 and x  I},
is A = B?
More details...
• U usually stands for the
universal set.
• We are usually given the
universal set. It is often a
set of numbers, such as
I (Integers), E (Even
Integers) or R (Reals).
More details...
• The complement of a set
(written A’ or Ā or ˜A )
contains everything in the
universal set that isn’t in
the set itself.
• If U = {a, b, c, … f} and
if A = {f, a, c, e}, find A’.
More details...
• Set A is a subset of B
( A  B) if and only if every
member of A is also in B.
• If B = {a, b, c, … f} and if
A = {f, a, c, e}, then A  B.
• Is A’  B ?
More details...
• Set A is a proper subset of B
( A  B) if and only if A  B
but A  B.
• A = {a, c, e}, B = {f, a, c, e} and
C = {a, c, e, f},
• Then A  B, and A  B.
• Moreover, B  C, but B  C.
More details...
• A set with no elements is
called the empty set and
is written as { } or  .
• The empty set is a subset
of all sets.
More details...
• The cardinality of a set is
the number of elements in
a set, and is written |S|.
• If K = {3, 5, 8}, then
|K| = 3
• If P is the set of all
subsets of K, find |P|.
More details...
• The cardinality of a set can
only be found if the set is
finite. But infinite sets
contain an infinite number
of elements.
More details...
• The union of A and B
(written AB) is the set
that contains all elements
of A and all elements of B.
More details...
• The intersection of A and B
(written AB) is the set
that contains all elements
that A and B have in common.
More details...
• If the intersection of A
and B is the empty set
(written AB =  ), then
A and B are disjoint sets.
More details...
• The difference set of A
and B (written A - B) is the
set that contains all the
elements in A that are not
in B.
More details...
• The set of elements that
are either in A or B, but
not both is called the
symmetric difference of A
and B and is written A  B.
More details...
• The Cartesian Product of
A and B, written A X B,
is the set of all ordered
pairs (a,b) where a  A
and b  B.
Example
• If L = {1, 3}, M = {2, 3, 4},
N = {3, 4}, O = {2, 4, 5},
and U = {1, 2, 3, 4, 5}
• Is N  M ?
• Is L  M?
• Is N  M?
Example
• If L = {1, 3}, M = {2, 3, 4},
N = {3, 4}, O = {2, 4, 5},
and U = {1, 2, 3, 4, 5}
• Write N’.
• Write N O.
• Write N  O.
Example
• If L = {1, 3}, M = {2, 3, 4},
N = {3, 4}, O = {2, 4, 5},
and U = {1, 2, 3, 4, 5}
• Write P(L).
• Write |O’|.
• Write M - O.
• Write M  O.
Example
• If L = {1, 3}, M = {2, 3, 4},
N = {3, 4}, O = {2, 4, 5},
and U = {1, 2, 3, 4, 5}
• Write L X N.
Example
A
B
AC  BD =
AC  BD =
AB  CD =
C
D
Example
A
B
C
F
CA  CF =
AD  CF =
AB  CF =
D
Any Questions?
• Time to practice now…