2. *
A set is a well-defined collection of distinct
objects. The objects in a set are called the
elements, or members of the set. If an
element a belongs to set A, then we write .
Otherwise, we write .

3. *
Example 1: Which of the following collections are
sets?
*The past presidents of the country
*Good players who have played for the national team
Example 2:
A = {1,2,3,4,5}
B = {x/x is an even number}
M = {5,10,15,…,75}
Q is a set of the presidents of the Republic of
the Philippines

4. *
Roster or Tabular Method
The roster method involves listing down the elements
of the set where a comma serves to distinguish each
distinct element of the set and enclosing them in
braces.
Examples:
A = {1,3,5,…}
B = {Monday, Tuesday, Wednesday, Thursday, Friday,
Saturday, Sunday}
P = {circles, triangles, squares, rectangles, spheres,
pentagons}
R = {2,4,6,…,20}

5. *
Rule Method or Defining Property Method or Set
Builder Notation
The rule method involves describing a set in terms of
its characteristic or property where only those
described or agreeing to that specific property are
considered elements of the set. It takes the form,
A = {x/”___”}
*Examples:
A = {x/x is a positive odd number}
B = {x/x is a day of the week}
C = {x/x is a letter in the English alphabet}
D = {x/3 < x < 10}

6. *
The cardinality of a set or the cardinal
number of a set is the number of elements
in the set A and is denoted by n(A).
A set is said to be finite if it is empty or if it
consists of exactly n elements where n is a
counting number. Otherwise it is an infinite
set.

7. *
*The set of all possible elements under
discussion is called Universal set and is
denoted by U.
*A set with no elements is called an empty
set or null set and is written as Ø or { }.

8. *
*Two sets A and B are equivalent if they
have the same number of elements, that is,
n(A) = n(B).
*Two sets A and B are equal if and only if
they contain exactly the same elements.
*Two sets are said to be disjoint if they have
no common elements.
*Two sets are overlapping or joint sets if
they have at least one element in common.

9. *
*A is a subset of B, denoted by , if
every element of A also belongs to B.
Find all subsets of A = {1,2,3}
*If and A ≠ B, then we say that A is a
proper subset of B and we write .
BA
BA
BA

10. *
*The power set of A is the set whose
elements are the subsets of A. it is denoted
by ρ(A) and the cardinal number is n(ρ(A)) =
2n(A).
Find the power set of A if A = {5, 10, 15}

11. *
*A Venn Diagram (named after Robert Venn)
is a geometric representation which
illustrates the relationships between and
among sets. It uses circles usually pictured
within a rectangle (universal set).
U
B
x
A
A B U
1
2 3
4 5
6

12. *
*The union of two sets A and B, denoted by
is the set containing all the elements
that belong to A or B
*Example: Find the union of the given pairs
of sets:
1. A = {4,5,6} ; B = Ø
2.
BA
}6,5,4,3{}3,2,1{

13. *
*The intersection of two sets A and B,
denoted by , is the set containing the
elements that are common to both A and B.
Example:
Find the intersection of the given sets
*A = {4,5,6} B = {4,8,10}
*X = {m,s,a} Y = {m,a,t.h.s}
*A = {a,f,g} B = {1,2,3,4}
BA

14. *
*The complement of a set A denoted by A’ is
the set of all elements in the universal set U
that are not found in A, that is
A’ = { .
Examples: Find the following sets:
*A’ if U = {a, b, c} and A = {c}
*B’ where U = {4,6,8,9} and B = {4,8}
}/ AxUx

15. *
*The difference of two sets A and B,
denoted by A – B, is the set of all elements
in B that are not in A.
Example: Find A – B.
*A = {w,x,y,z} B = {x,z}
*A = {a,b,c,d} B = { }
*A = {2,3,4,5,6} B = {1,3,5}

16. *
*The symmetric difference of two sets A and
B denoted by A – B is the set
A – B = {x/ and }.
*The Cartesian product of A and B denoted
by A x B is the set of all ordered pairs (x,y)
where and , i.e.
A x B = {(x,y)/ and }.
BAx BAx
Ax Bx
Ax Bx