Introduction to Set theory

INTRODUCTION TO
SET THEORY
Prepared by:
S herwin V. Labadan BS E D-2

S E T THE OR Y
19
is a branch of mathematics that
studies sets or the mathematical
science of the inifite.
It studies properties of sets and
help people to organize things into
groups.

1 C orinthians
14:40
20
"B ut all things must be
done properly and in
order."

22
is any well-defined collection of
“objects.”
are the objects in the set.
of the sets

A = {a, b, c, d, e, f,...}
Sets elements
E xamples of
S ets:
BS E D-Math female= {Welcez, R onah, Julie,
R oxanne, C harmaine}C olors ofa rainbow= {red, orange, yellow, green, blue,
indigo, violet}
S tate ofmatter={solid, liquid, gas, plasma}
A={x lx is a positive integer less
than 10}
B={x lx is a setofvowel letters}

B={x lx is a setofvowel
letters}
B={a, e, i, o,
u}
a
B
 b B
an object is an
element of a set
an object is NOT
an element of a set

ME THODS OF
WR ITING S E TS
25
“Tabulation
Method”
“Set Builder
Notation”
-the elements of
the sets are
enumerated
-a descriptive
phrase
{x l P(x)}
B={a,e,i,o,u} B={x lx is a setofvowel lette
E xampl
e:
E xampl
e:

A={x lx is a positive integer
less than 10}
A={1,2,3,4,5,6,7,8,9}
C ={xlx is a letter in the word dirtC ={d, i, r, t}
D={xlx is odd and x>2}D={3,5,7,9,11,... }

}F INITE S E TS
setwhose
elements are
limited or
countable

}
INF INITE
S E TSsetwhose
elements are
unlimited or
uncountable

UNIT
SET
setwith one one
element“Singleton”
E MP T Y
S E T“”Null Set”
or { }
setwith no
elements
setofall elements currently
under considerationUNIVE R S AL
S E T U or 
------------
--------------
--------

EXERCISE 1
1. P={xl x is a whole number greater than 1 but
less than 3}2. M={xl x is an integer less than 2 butgreater
than 1}3. S ={xl x is the setofpositive integers less
than zero}
 4. U={3,6,9,12,15,18,21,24,27}
5. Q={Xia}
6. L={xl x is a vowel letter ofthe word
rhythm }

C AR DINALITY
31
T he cardinalnumber of a set is the
number or elements or members
in the set
n(A)

The cardinal
number of A is 9 or
n (A)= 9.
{1,2,3,4,5,6,7,8,9}
T he cardinal
number of C is 4
or n (C )= 4.
T he cardinal
number of B is 5 or
n (B )= 5.

EXERCISE 2
1. 
2. Q={1,2,3}
3. I={ }
4. V= {{1,2,3},{4,5}}
5. H={ ,{a},{b}, {a,b}}

VENN
DIAGRAM“Set Diagram”
- a picturial presentation
ofrelation and
operations on sets
JOHN VENN
(1834-1923)

S UBS E T
35
IfA and B are sets, A is called
subsetofB
A B
ifand only if, every elementofA is
also an elementofB.
BxAxxBA  ,

A
B
U
a
b
cd
e
f
g
A={c,d,e}
U={a,b,c,d,e,f,g}
B={a,b,c,d,e}
A B

PR OPE R
S UBS E T
37
IfA and B are sets, A is a proper
subsetofB
ifand only if, every elementofA is in B,
butthere is atleastone elementofB
thatis notin A.
A B

A
B
U
a
b
cd
e
f
g
A={c,d,e}
U={a,b,c,d,e,f,g}
B={a,b,c,d,e}
C
C ={a,b,c,d,e}
BA
CA
BA
BC 

E QUAL S E T
39
G iven setA and B, A equals B
ifand only if, every elementofA is in B,
and every elementofB is in A.
BA
ABBABA 

A
B
U
a
b
cd
e
f
g
A={c,d,e}
U={a,b,c,d,e,f,g}
B={a,b,c,d,e}
C
C ={a,b,c,d,e}
CB 
BC 
CB 

POWE R S E T
41
G iven a setS from universe U, the
power setofS
is the collection (or sets) ofall subsets
ofS .
)(SP
2
n

1. A= {e, f} as it is a
subset of all
sets
2. S ={0, 1, 2}
}}2,1,0{},2,1{},2,0{},1,0{},2{},1{},0{,{)( SP
}},{},{},{,{)( fefeAP 
2)( An 4)( AP
3)( Sn 8)( SP
}{)(.3 P
0)( n 1)( P

UNION
44
The union ofA and B,
is the setofall elements ofx in U
such thatx is in A or x is in B.
}{ BxAxlxBA 
BA

INTE R S E C TIO
N
45
The intersection ofA and B,
such thatx is in A and x is in B.
}{ BxAxlxBA 
BA

DIS JOINT S E TS
46
 BA
Two sets are called disjoint(or non-
intersecting) ifand only if, they have no
elements in common

C OMPLE ME NT
47
The complementofA (or absolute
complementofA)
such thatx is notin A
}{' AUlxxA 
'A

DIF F E R E NC E
48
The diffference ofA and B
(or relative complementofA and B)
such thatx is in A and x is notin B.
'{~ BABxAxlxBA 
BA~

S YMME TR IC
DIF F E R E NC E
49
their symmetric difference as the set
containing ofall elements thatbelong to A or
to B, butnotto both and B
)}()({ BAxBAxlxBA 
BA
IfsetA and B are two sets
)(~)()'()( BABAorBABAor 

OR DE R E D PAIR
50
In the ordered pair,
),( ba
a is called the firstcomponentand
b is the second component
),(),( abba 

EXERCISE 3
1. (2,5)=(9-7,
2+3)
2. {2,5} ≠ {5,2}
3. (2,5) ≠ (5,2)
TR UE OR
F ALS E

C AR TE S IAN
PR ODUC T
52
The C artesian productofsets A and B,
AxB
}),{( BAandblabaAxB 
is

F ind the cartesian productofa
given set:
A={2,3,5} and B={7,8}
1. AxB=
{(2,7),(2,8),(3,7),(3,8),(5,7),(5,8)}
2. BxA={(7,2),(8,2),(7,3),(8,3),(7,5),(8,5)}
3. AxA=
{(2,2),(2,3),(2,5),(3,2),(3,3),(3,5),(5,2),(5,3),(5,5)}
EXAMPLE

Identify the elements of a given
kind of set and its cardinality
1. A= {xlx is a consonant letter
of
s upercalifragilis ticexpialidoci
ous }
2. B = {xlx is a squares of the
first five counting number}
3. S et C contains all the
factors of 244. D= {xlx is an positive integer, -
QUIZ

L et U= {postive integers from
1 to 10}
A= {1,2,4,6,8,10}
B = {1,3,5,7,8,9}
C = {1,2,3}
D= {2,3,4}
E = {3,4,5}
F = {4,5,6}
F IND:
1.
2.
3.
4.
5.
'' BA
FED 
EC
DB
EA

ANSWER KEY
S tation 1: Null S et
S tation 2: Proper
S ubset
S tation 3:
C ardinality
S tation 4: S ubset
S tation 5: R ule
Method
S tation 6:
Difference
S tation 7:
Universal S et
S tation 8:
C omplement
S tation 9:
S tation 11: E lement
S tation 12: C artesian
Product
S tation 13: Union
S tation 14: Power S et
S tation 15: Universal S et
E XE R C IS E 1:
1. Unitset
2. E mpty set
3. E mpty set
4. Universal set
5. Unitset
6. E mpty set

E XE R C IS E 2:
1. 0
2. n( Q)=3
3. n(I)=1
4. n(V)=2
5. n(H)=4
E XE R C IS E 3:
1. TR UE
2. F ALS E
3. TR UE
QUIZ:
1. A={s,u,p,e,r,c,a,l,i,f,g,t,x,d,o,u}
n(A)=16
2. B={1,4,9,16,25}
n(B)=5
3. C ={1, 2, 3, 4, 6, 8, 12, 24}
n(C )=8
4. D={1,2,3,4,5,6,7,8,9}
n(D)=9
1. {2,3,4,5,6,7,8,9,10}
2. {4}
3. {1,2,4,5}
4. {1,5,7,8,9}
5. {1,2,3,4,5,6,8, 10}

Thank you
for sliding!
@ S ecretC LS he
rwin
S herwin Barcelona Villa
Labadan
slabadan8@ gmail.com
+F ollow me
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