Hg
Name: Class 11 Date:
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
Sets
• A set is a well-defined collection of objects. Order of objects is immaterial. Sets are represented either by Roster
form or Set Builder form.
• Roster or Tabular form: In this form, all the elements of a set are listed, the elements are being separated
by commas and are enclosed within braces { }. Set doesn’t change if any of its elements are repeated, that’s
why we generally don’t repeat elements describing a set.
• Set builder form: In this form, all the elements of a set possess a single common property which is not possessed
by any element outside the set. Syntax: { x | characterstic property that x follows}
• Cardinality of a Set: Cardinality of a set A is the number of elements in A and is denoted by n(A) or |A|.
• Empty or Null Set: A set which does not contain any element is called empty set. It is denoted by { } or the
symbol Ø. Clearly, n(Ø) = 0.
• Singleton Set: If n(A) = 1, then A is called a singleton set.
• Finite & Infinite Sets: A set which consists of a definite number of elements is called finite set, otherwise, the set
is called infinite set.
• Equal Sets: Two sets A and B are said to be equal if they have exactly the same elements.
A = B, if x Є A x Є B and x Є B x Є A
• Subset: A set A is said to be subset of a set B, if A has some or all elements of B and is denoted by A ⊆ B.
Intervals are subsets of R.
A ⊆ B, if x Є A x Є B
• Proper Subset: A set A is said to be a proper subset of B, if A has some elements of B and is denoted by A ⊂ B.
o Ø ⊆ All sets
o Ø ⊂ All sets except Ø
o A ⊄ A
o A = B, if A ⊂ B and B ⊂A.
• Super-set: A set B is said to be super-set of a set A, if A ⊆ B and is denoted by B ⊇ A.
• Power Set: A power set of a set A is collection of all subsets of A, is denoted by P(A). If n(A) = k, then n(P(A)) = 2
k
.
Operations on Sets:
• Union: The union of 2 sets A & B is the set of all elements which are either in A or in B and is denoted by A B.
A B = { x | x ∈ A or x ∈ B }
• Intersection: The intersection of 2 sets A & B is the set of all elements which are common, is denoted by A B.
A B = { x | x ∈ A and x ∈ B }
• Difference: The difference of two sets A & B in this order is the set of elements which belong to A but not to B
and is denoted by A – B.
A – B = { x | x ∈ A and x ∉ B }
Properties of Union and Intersection:
• A B = B A; A B = B A (Commutative law)
• ( A B ) C = A ( B C); ( A B ) C = A ( B C ) (Associative law )
• A Ø = A; A Ø = A (Law of Ø, Ø is the identity of )
• A A = A; A A = A (Idempotent law)
• U A = U; U A = A (Law of U)
• A ( B C ) = ( A B ) ( A C ); A ( B C ) = ( A B ) ( A C ) (Distributive Law)
• Disjoint Sets: Two sets are said to be disjoint if A B = Ø.
• Universal Set: The biggest set of all sets in a specific problem domain is termed as the universal set. It is denoted
by letter U.
Hg
Name: Class 11 Date:
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
• Complement: The complement of a subset A of universal set U is the set of all elements of U which are not the
elements of A and is denoted by A’ or A
c
.
• De Morgan’s Law: For any two sets A and B, (A B) = A B and ( A B ) = A B
• If A and B are finite sets such that A B = Ø, then n (A B) = n (A) + n (B).
• If A B Ø, then n (A B) = n (A − B) + n (A B) + n (B − ) = n (A) + n (B) – n (A B)
• For finite sets A, B, C:
n (A B C) = n (A) + n (B) + n (C) – n (A B) – n (B C) – n (C A) + n (A B C)
• Open and Closed Intervals: An interval that includes boundary elements is closed, otherwise, its open. Open
interval is denoted by parenthesis ( ). Closed interval is denoted by square brackets [ ].
e.g. (2, 6) means 2 < x < 6, [2, 6] means 2 ≤ x ≤ 6, (2, 6] means 2 < x ≤ 6 and [2, 6) means 2 ≤ x < 6
• Venn Diagrams: Relationships between sets can be represented by means of diagrams known as Venn diagrams.
They consist of rectangles and closed curves - usually circles. The universal set is represented usually by a
rectangle and its subsets by circles.
• Number Sets:
Symbol
Name of the set
Examples in Roster Form
Few relationships in
set terminology
C
Complex Numbers
{…,2 + 5i, 2, i, ….}
C = A A
c
A
Algebraic Numbers
{x | x = Complex Roots of all non-zero polynomials in
1 variable with rational coefficients}.
{…..,φ, -2, 1, ½, 2, 1- i, 2+3i ….}
A ⊂ C
A
c
Transcendental Numbers
{…, , e, + i…}
A
c
⊂ C
R
Real Numbers
{….,-2, -1, 0.33.., 0, ½, ,…}
R ⊂ C
A
R
Real Algebraic
{….,-2, -1, 0.33.., 0, ½, 2, φ,…}
A
R
⊂ R
N
Natural Numbers
{1,2,3,...} or {0,1,2,3,...}
N ⊂ Z, N ⊂ A
Z
Integers
{..., -3, -2, -1, 0, 1, 2, 3, ...}
N ⊂ Z ⊂ R, Z ⊂ A
Q
Rational Numbers
{….,-1, -2, 1, 2, 0, ½, 2/3…}
N ⊂ Z ⊂ Q ⊂ R, Q ⊂ A
T
Irrational Numbers
{….,, 2, φ, 1.32301001…,…..}
T = (R – Q) ⊂ R ⊂ C
I
Imaginary Numbers
{….,-2i, 3i, i…}
I ⊂ C
Z+
Positive Integers
{….,2, 3…}
Z+ ⊂ Z
R+
Positive Real Numbers
{….,2, 1, 0.33.., 0, ½, ,…}
R+ ⊂ R
Q+
Positive Rational Numbers
{1, 2, 0, ½, 2/3, 1.5,…}
Q+ ⊂ Q
