SET Example :

- is a collection of objects called 𝐸 = {xx is a collection of vowel letters}

members or elements .
The objects are called the elements or Write the ff. Sets in Roster Form
members of the set.

𝜖 = element of a set 1. A= {xx is the letter of the word

discrete}
 = not an element of a set.

Example :
2. C = {xx is the set of zodiac signs}
1. 𝐸 = {xx is a set of vowel letters}
“Read as Set E equals the set of all x such
that x is a set of vowel letters”
TRY THIS :
2. 𝐴 = {xx is a letter in the word Write the ff. Sets in Rule Form
mathematics.}
1. D = {Narra, Mohagany, Molave, …}

3. B = {xx is a positive integer,

3  x  8.} 2. F = {Botany, Biology, Chemistry,
Physics, …}

Methods of Writing Sets

1. Roster Method. The elements of the set Some Terms on Sets

are enumerated and separated by a
comma it is also called tabulation method. 1. Finite set
- is a set whose elements are limited or
Example : countable, and the last element can be
identified.
E = {a, e, i, o, u}
Example :

2. Rule Method. A = {xx is a positive integer less than 10}

- A descriptive phrase is used to
describe the elements or members of the C = {d, i, r, t}
set it is also called set builder notation,
symbol it is written as
{𝑥|P(𝑥)} .

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2. Infinite set Example :
- is a set whose elements are unlimited
or uncountable, and the last element U = {xx is a positive integer, x2 = 4}
cannot be specified U = {1, 2, 3,…,100}

Example :
6. Cardinality
F = {…, –2, –1, 0, 1, 2,…} - The cardinal number of a set is the
G = {xx is a set of whole numbers} number of elements or members in the
set, the cardinality of set A is denoted by
n(A).
3. Unit set
- is a set with only one element it is also Example: Determine its cardinality of the
called singleton. ff. sets

E = {a, e, i, o, u}
Example :
I = {xx is a whole number greater
than 1 but less than 3}
J = {w} A = {xx is a positive integer less than 10}

4. Empty set
- is a unique set with no elements (or
null set), it is denoted by the symbol 
or { }. Power Set
Given a set S from universe U, the
Example : power set of S denoted by 𝑝(𝑠), is the
collection (or sets) of all subsets of S.
L = {xx is an integer less than 2 but
Determine the power set of
greater than 1}
a) A = { e, f },
No integer less than 2 but greater than 1 b) B = {1, 2, 3}.
exist
A = {e, f}
M = {xx is a dinosaur bear in Manila Zoo}
There is no dinosaur bear in Manila Zoo
B = {1, 2, 3}
5. Universal set
- is the all sets under investigation in
any application of set theory are assumed
to be contained in some large fixed set,
denoted by the symbol U.

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Venn diagram 2. Intersections of Sets
- is a drawing in which geometric figures - the largest set which contains all the
such as circles and rectangles are used to elements that are common to both the
represent sets. One use of Venn diagrams sets.
is to illustrate the effects of set operations. - The symbol ∩ is employed to denoted
A Venn diagram uses circles that overlap by the symbol ∩
or don't overlap to show the - Read as “A intersection B” or “the
commonalities and differences among intersection of A and B
things or groups of things.
Example :
Operations of Sets A = { 1, 2, 3, 4, 5 6}
B = { 1, 3, 5, 7, 9 }
1. Union of Sets
A ∩ B = { 1, 3, 5, }
- the smallest set which contains all the
elements of both the sets.
- Denoted by the symbol ∪
- Read as “A union B” or “the union
of A and B

Example : 𝐴𝑈𝐵
A = { 1, 2, 3, 4, 5 6}
B = { 1, 3, 5, 7, 9 }
A ∪ B = { 1, 2, 3, 4, 5, 6, 7 ,9 }

Taking every element of both the sets A

and B, without repeating any element, we
get a new set. 3. Complement of Sets
- the set of elements that are members
of the universal set U but are not in A ∪ B.
It is denoted by (A ∪ B ) ’

Example :

Given :

U = {a, b, c, d, e, f , g , h , i, j }
A = {a, b, c, d}
B = {d, h, i, j }
A ∪ 𝐵 = {𝑎, 𝑏, 𝑐, 𝑑, ℎ, 𝑖, 𝑗 }
(𝐴 𝑈 𝐵)′ = {𝑒, 𝑓 , 𝑔}

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 d) 𝑄 − 𝑃 means elements of Q which are
not the elements of R

5. Cartesian Product of sets

- the multiplication of two sets to form
the set of all ordered pairs. The first
4. Difference of two sets element of the ordered pair belong to first
- If A and B are two sets, then their set and second pair belong the second
difference is given by A - B or B - A. set.
- A - B means elements of A which are
not the elements of B. Example :
- B - A means elements of B which are
not the elements of A. 𝐴 = {𝑚𝑖𝑙𝑘, 𝑠𝑜𝑑𝑎 , 𝑡𝑒𝑎}
B = {𝑝𝑎𝑠𝑡𝑎 , 𝑐𝑎𝑘𝑒 }
Example :

𝑃 = {10, 11, 12, 13, 14, 15 , 16 } 𝐴𝑥𝐵=

𝑄 = { 10, 12, 14, 16, 18}
𝑅 = {7, 9, 11,14, 20 }

Find the following : 𝐵𝑥𝐴 =

a) 𝑃−𝑄
b) 𝑄−𝑅
c) 𝑅−𝑃 A = { cat , dog }
d) 𝑄−𝑃 B = {fish , bird , turtle }

Answers:
AxB =
a) 𝑃 − 𝑄 means elements of P which are
not the elements of Q.

BxA =
b) 𝑄 − 𝑅 means elements of Q which are
not the elements of R

c) 𝑅 − 𝑃 means elements of R which are

not the elements of P

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Venn Diagram in case of two sets Let S = soda
T = tea

Where :
A = number of elements that belong to set
Venn Diagram in case of three sets
X only
B = number of elements that belong to set
Y only
C = number of elements that belong to
both set X and Y (X ∩ Y)
Z = number of elements that belong to
none of the sets X and Y

From the above figure, it is clear that

n(X) = A + C ; Where :
n (Y) = B + C ;
n(X ∩ Y) = C; A = number of elements that belong to set
n ( X ∪ Y) = A +B+ C X only
B = number of elements that belong to set
Total number of elements = A+ B + C+ Z Y only
C = number of elements that belong to set
Example : W only
D = number of elements that belong to
In a party, 150 visitors are randomly selected. set X, Y and W
40 like soda , 50 like tea and 35 like both soda E = number of elements that belong to
and tea . set X and Y not W
F = number of elements that belong to
A) How many students like only soda? set X and W not Y
B ) How many students like only tea? G = number of elements that belong to
C) How many students like neither soda set Y and W not X
nor tea?
D) How many students like only one of Z = number of elements that belong to
soda or tea? none of the sets X , Y and W
E ) How many students like at least one of
the beverages?

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From the above figure, it is clear that

n(X) = A + D + E + F ;
n (Y) = B + E + D + G ;
n (W) = C + D + F + G

n (X ∩ Y) = D + E ;
n ( X ∪ Y) = A +B+ D + E + C + G :

n (X ∩ W) = F + D;
n ( X ∪ W) = A + C + D + E + F + G :

n (Y ∩ W) = D + G
n ( Y ∪ W) = B + C + D + F + G

Total number of elements = A+ B + C+ D

+E+F+G+Z

Example :

Let B = students course is BIO

E = students course is ENG

100 students were interviewed

28 took PE, 31 took BIO, 42 took ENG, 9
took PE and BIO, 10 took PE and ENG, 6
took BIO and ENG, 4 took all three
subjects.

a) How many students took none of the

three subjects?
b) How many students took PE but not
BIO or ENG?
c) How many students took BIO and PE
but not ENG?

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SEATWORK :

Suppose 135 customer in a café are

surveyed about their favorite desserts .
Draw a Venn diagram box based on the
following statements and then use it to
answer the following questions .

63 like leche plan

61 like cake
53 like ice cream
28 like cake and ice cream
31 like leche plan and cake
22 like leche plan and ice cream
15 like all these three flavors

1. How many customers do not like any

of the green desserts ?
2. How many customers who like only
cake ?
3. How many customers who likes leche
plan and cake but not ice cream ?
4. How many customers who likes leche
plan and ice cream but not cake ?
5. How many customers who likes only
at least one dessert ?
6. How many customers who likes cake
and ice cream but not leche plan ?

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