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Accelerated III
(Acc. Precalculus)
August 20, 2015
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 a set?
• A set is a well defined
collection of objects.
• No, S = {some numbers} is
not well defined.
• W = {Fred, George, Ron} is
a set.
What is in a set?
• The objects in a set are
called elements or
members.
•  means “is a member of”.
• We can write 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  Z}
• or by roster
S = {1, 2, 3, 4, …,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  Z},
is A = B?
• Yes, 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 Z,
W or R.
More details...
• The complement of a set
(written A’ or Ā ) 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...
• If U = {a, b, c, … f} is the
universal set and
• if A = {f, a, c, e},
• The complement of set A
• which is written 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, A’ = {b, d}, so 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.
• Furthermore, B  C,
but B  C.
More details...
• A set with no elements is
called the empty set or
null set and is written as {}
or  .
• The empty set is a subset
of all sets.
More details...
• The Power set of a set is
the set of subsets of the
set.
• If A = {3, 5}, then P(A) =
• { {3}, {5}, {3, 5}, }
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 |P(K)|?
• |P(K)| = 8
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
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.
• So {1, 2} X {6, 8} =
• {(1, 6), (1, 8), (2, 6), (2, 8)}
Work These!
• If L = {1, 3}, M = {2, 3, 4},
N = {3, 4}, O = {2, 4, 5}, and
U = {1, 2, 3, 4, 5}
• 1. Is N  M ?
• 2. Is L  M?
• 3. Is N  M?
Work These!
• If L = {1, 3}, M = {2, 3, 4},
N = {3, 4}, O = {2, 4, 5}, and
U = {1, 2, 3, 4, 5}
• 4. Write N’.
• 5. Write N O.
• 6. Write N  O.
Work These!
• If L = {1, 3}, M = {2, 3, 4},
N = {3, 4}, O = {2, 4, 5}, and
U = {1, 2, 3, 4, 5}
• 7. Write P(L).
• 8. Write |O’|.
• 9. Write M - O.
• 10. Write M  O.
Work These!
• If L = {1, 3}, M = {2, 3, 4},
N = {3, 4}, O = {2, 4, 5}, and
U = {1, 2, 3, 4, 5}
• 11. Write L X N.
Any Questions?