Download Sets #1

Honors Geometry
March 21, 2016
Sets
What is a set?
• A set is a well defined
collection of objects.
• Is S = {some numbers} a
set? no
• Is W = {Fred, George, Ron}
a set? yes
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?
• Sets can be written by rule
S = {x| 0 < x < 10 and
x  Integers}
• or by roster
S = {2, 3, 4, 5, …,9}
How do you write a
set?
• They are enclosed by
brackets and the elements
are separated by commas.
The series of dots (… ) are
called ellipses and mean
“the pattern continues”.
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? yes
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 Z =
integers, R = Reals, etc.
More details...
• The complement of a set
(written A’, Ā, ~A 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’.
• A’ = {b, d}
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 ?
• Yes
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.
• Furthermore, 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
• What is |{2, A, 5k, 3}|?
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.
• If P = {1, 2, 3, 4} and Q =
{3, 4, 5, 6}, find PQ.
• PQ ={1, 2, 3, 4, 5, 6}
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.
• If P = {1, 2, 3, 4} and Q = {3,
4, 5, 6}, find P  Q.
• P  Q ={3, 4}
More details...
• If the intersection of A
and B is the empty set
(written AB =  ), then A
and B are disjoint sets.
• Give an example of disjoint
sets.
More details...
• The difference set of A
and B (written A - B) is the
set that contains all
elements in A that are not
in B.
• If P = {1, 2, 3, 4} and Q =
{3, 4, 5, 6}, find P - Q.
• P – Q = {1, 2}
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.
• If P = {1, 2, 3, 4} and Q =
{3, 4, 5, 6}, find P  Q.
• P  Q = {1, 2, 5, 6}
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}
• Is N  M ? Yes
• Is L  M? No
• Is N  M? Yes
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 N’. {1,2, 5}
• Write N O. {4}
• Write N  O. {2, 3, 4, 5}
Example
• If L = {1, 3}, M = {2, 3, 4},
N = {3, 4}, O = {2, 4, 5}, and
U = {1, 2, 3, 4, 5}
• Write all the subsets of 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 all the subsets of L.
• Write |O’|. 2
• Write M - O. {3}
• Write M  O. {3, 5}
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.
• L X N = {(1, 2), (1, 3), (1, 4),
(3, 2), (3, 3), (3, 4)}
Any Questions?