M102 : Single Variable Analysis
Abhijit Mookerjee
Senior Professor and Dean (Faculty)
S.N. Bose National Centre for Basic Sciences
Kolkata
Visiting Professor, IISER, Kolkata
Phone : 2335 5706 Mobile : 9830335860
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Reference Books
Tutors
Prof. Binayak Dutta-Roy Prof. A. Mookerjee
- Understanding Mathematics : K.B. Sinha R.L. Karandikar C. Musili S.
Pattanayak D. Singh A. Dey : Universities Press
- Calculus : T.M. Apostol Vol 1 : John Wiley and Sons
- Mathematics, its contents, methods and meaning : Alexandrov, Kolmogorov,
Laventsev
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Mathematics of Sets
•The notion of a set is intuitive. It is a primitive notion. We shall not
formally define it.
• A set is a collection of things, objects or even ideas :
e.g. a flock of birds, a tribe of Santhals, a crowd of college students,a
cricket team, an electrorate.
•We can always define who/what belongs to the set and who/what
does not.
Flock ⇒
B = {crows, peacocks, parrots, vultures . . .} ⇒ defn by extension
B = {x : x = a species of bird} ⇒ defn by intention
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Mathematics of Sets
A pigeon belongs to the set, but a tiger does not
x = pigeon ∈ B x = tiger 6∈ B
• This partition of everything into what belongs to the set and what
does not is the nearest to a formal definition of that set
Take another example :
Integers Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .} or
Z = {n : n = integers}
T = {t : t = triangles with area 1 unit2 }
E = {e : e = all Indian nationals with age > 18 yrs}
S = {− 21 , 0, 12 , 32 }
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Mathematics of Sets
Order is immaterial in a set.
Nor is the number of times the object within a set appears in the
extensive definition.
For example :
- The sets {2, 4, 6, 8}, {4, 2, 8, 6} and {2, 2, 4, 6, 6, 6, 8} are identical
- The sets {g, a, n, g, a}, and {n, a, g} are identical
- The sets {4, 5, C, B} and {B, 4, C, 5, C} are identical
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Mathematics of Sets
•A set A is called a subset of another set B if : x ∈ A implies x ∈ B. We
write this as
A⊆B if x∈A⇒x∈B
•Two sets are called equal if :
A ⊆ B and B ⊆ A ⇒ A = B
•A set A is called a proper subset of another set B if :
A ⊂ B and A 6= B
Examples,
Z = {n : n = integer} , Z + = {n : n = positive integer} , R = {x : x = real number}
Z+ ⊂ Z and Z⊂R
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Mathematics of Sets
Concept of a null set {∅}
It is a set with no elements ! Is it a set ?
It is required for logical completeness
{∅} = {t: t=triangles sum of whose angles > 2π}
{∅} = {m:m=mammals born through parthenogenesis}
{∅} = {n:n=no of quantum states with > 1 fermion occupancy}
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Mathematics of Sets : Important Logical
Statements
X ⊆X
{∅} ⊆ every set X
X =Y ⇐⇒ X ⊂ Y and Y ⊂ X
X = Y =⇒ (x ∈ X ⇐⇒ x ∈Y)
X =X
X =Y =⇒ Y=X
X =Y, Y=Z =⇒ X =Z
X ⊂ Y, Y⊂ Z =⇒ X ⊂Z
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Mathematics of Sets : Some definitions
Union : X ∪ Y = {x: x ∈ X or x ∈ Y}
X ∪Y =Y ∪X
(X ∪ Y) ∪ Z = X ∪ (Y ∪ Z)
X ⊂ Y ⇐⇒ X ∪ Y = Y
Intersection : X ∩ Y = {x: x ∈ X and x ∈ Y}
X ∩ Y =Y ∩ X
(X ∩ Y) ∩ Z = X ∩ (Y ∩ Z)
X ∩ Y ⊂ X, X ∩ Y ⊂ Y
X ⊂ Y ⇐⇒ X ∩ Y = X
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Mathematics of Sets : Some definitions
Difference : X \ Y = { x: x=x∈ X and x6∈ Y }
Symmetric Difference : X 4 Y = (X \ Y) ∪ (Y \ X )
Power Set : P((X)) = { A : A ⊂ X }
When we define a set, say N I = {x: x=”non-Inidans”} , we have to
further specify in what context this set is defined. E.g. are we talking
about non-Indians who are living in Kolkata or non-Indians who live
in India or non-Indians who are members of humanity. This big set U
such that N I ⊂ U is called the universal set of our problem.
Complement X c = U \ X
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De Morgan’s Laws
- (X ∪ Y)c = X c ∩ Y c
- (X ∩ Y)c = X c ∪ Y c
- A4B = (A ∩ Bc ) ∪ (B ∩ Ac )
Example : Let U = {x: x=students in this class}. X = { x: x=students in this
class who are good in physics}, and Y = { x: x=students in this class who
are good in maths}. X ∪ Y = {x: x=students in this class who are good in
physics or maths or both}
(X ∪ Y)c = {x: x=students in this class who are not good in physics and
maths}
Since X c are the students in this class who are not good in physics and
Y c are those who are not good in maths, X c ∩ Y c are those who are
not good in physics and maths. One of the De Morgan’s Law is verified.
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