SET INTERVAL ON THE NUMBER LINE
1. Let A = 
and B={x∈IR:-7< x ≤ 3}Represents these set intervals on two separate number lines
Solutions
For A = 
For B = 
Examples
Using the sets A and B defined above, state and represents the following sets on same number line
a) A 
B b) A′ c) B′ d) A U B′
Solutions
a) A 
B
A 
B = 
b) A′
A′ = 
c)B′
B′ = 
a)
(d)A U B′
A U B′ =
QUESTION
i) Represent the above sets on one number line
ii) Draw and state each of the following sets on separate number lines
a) A ∩ B b) A ∪B c) B′ d) A∩B′
Solution
(i)
(ii)(a) A
b) A U B
c) B′
QUESTIONS.
1. Represents and then draw on one number line the following set interval
Using the above set interval, represent and state the following
i) A 
B ii) A 
C iii) C 
B‘ iv) (A
B) 
C
VENN DIAGRAMS
Sets can be represented in the form of diagrams called Venn diagrams
– The universal set is represented by a rectangle
– Subsets of U are represented by a circle in universal set
Uses of Venn diagram
i) To illustrate sets identity
ii) To find number of members in a given set
1. Illustration of set identity
Example; Illustrate by use of Venn diagram (A U B) 
A = A
Solution.
Two different methods can be used
i) Shading method
ii) Numbering of disjoint subsets
i) Shading method, i.e. to show (A 
B) ∩ A = A
L. H. S → (A 
B) ∩ A
Shade (A 
B) by vertical lines
Shade (A 
B) 
A by horizontal lines
Now (A 
B) 
A = region shaded
= A
= R. H. S
∴ (A 
B) 
A = A
ii) Numbering of disjoint
Solutions
L. H. S = (A 
B) 
A
Now A 
B = subsets 1, 2, 3
But A = sub 1, 2
(A 
B) 
A = subsets 1, 2
=A
= R. H. S
Example
Use Venn diagram to show A
(B 
C) = (A 
B) 
(A 
C)
Solution
L. H. S = A U (B 
C)
Now B 
C 
subsets 5, 6
A U (B 
C)
Subsets 1, 2, 5, 4 and 6
R. H. S = (A U B) 
(A U C)
A U B
subsets 1, 2, 3, 4, 5, 6
A U C 
subsets 1, 2, 3, 4, 5, 6, 7
(A U B) ∩ (A U C) = 1, 2, 5, 4, 6
A
(B 
C) = (A
B) 
(A 
C)
QUESTION
Use a Venn diagram to show the following
i) (A 
B) 
A = A
ii) A
(B 
C) = (A 
B) 
(A 
C)
LAWS OF ALGEBRA OF SETS
Set operations obey the following laws
1. Commutative laws
A U B = B U A
A 
B = B 
A
2. Associative laws
a) (A U B) U C = A U (B U C)
b) (A 
B) 
C = A 
(B 
C)
3. Distributive laws
a) A U (B 
C) = (A U B) 
(A U C)
b) A 
(B U C) = (A 
B) 
(A 
C)
4. De -Morgan’s laws
a) (A U B)′ = A′ 
B′
b) (A 
B)′ = A′U B′
5. Identity laws
a) A 
µ = µ
b) A 
µ = A
c) A 
Φ = A
d) A 
Φ =Φ
e) A\Φ = A
f) A\A = Φ
Examples
Use laws of algebra of set to simplify
1. (A
(A 
B)′)′
Solution
(A 
(A 
B)′)′ ≡(A
(A′ 
B′))′ De-Morgan’s law
≡((A 
A′)
B′ )′Associative law
≡ (Φ 
B′) Complement law
≡ (Φ)′Identity law
≡ µ complement law
(A
(A U B)′)′ = µ
Examples
Use the laws of algebra of sets to prove
(A
(B 
C′)) 
C = (A 
C) 
(B 
C)
Solution
L.H.S (A
(B 
C′)) 
C
= (((A 
B) C′)
C…….. Associative law
=((A 
B) U C) 
(C′ 
C) ………distributive law
= ((A 
B) 
C) 
(µ) …………complement law
= (A 
B) 
C……………. identity law
= (A 
C) 
(B 
C) ……………distributive law
= R. H. S
Exercise
1. Use laws of algebra of set to simply
i) (A 
B) 
(A 
B’)
ii) (A’ 
B’) 
(A 
B)
iii) (A 
B) U (A – B)
iv) A 
(A 
B)
2. Use laws of algebra to prove
i) (Z 
W)′ 
W = Φ
ii) (X 
Y’) 
(X 
Y) 
(Y 
X′) = X 
Y
iii) (A – B) 
A = A
Note
A – B = A 
B′ by definition
Number of elements in a set
The number of elements in set A is denoted by n (A)
Example
Let A be a set of all positive odd integers which are less than 10. Find n (A)
Solution
A = {1, 3, 5, 7, 9}
Now n (A) = 5
Examples
Let A ={x ∈ IR:x2-x-2=0}. Find n (A)
Solution
Note
i) The number of elements of a set is defined only for a finite set
ii) If A 
U then the number of elements of A′ is n(A′) = n(µ) – n(A)
Example
If A 
U and B 
U then show that n (A 
B) = n(A) + n(B) – n(A 
B)
Proof
Refer to the Venn diagram below
Represents the number of elements in disjoint subset as follows
Let n (A 
B′) = a n (A′ 
B) = c
n (A 
B) = b
R. H. S = n (A) + n (B) – n (A 
B)
= (a + b) + (b + c) – b
= a + 2b + c – b
= a + b + c
n (A 
B)
L. H. S