SETS AND RELATIONS
∗ What is Set ?
Set : A set is a collection of well defined objects which are distinct from each other
Set are generally denoted by capital letters A, B, C, .... etc. and the elements of the set by a, b, c .... etc.
If a is an element of a set A, then we write a ∊A and say a belongs to A.
If a does not belong to A then we write a ∉ A,
e.g. The collection of first five prime natural numbers is a set containing the elements 2, 3, 5, 7, 11.
Ex. Which of the following are sets? Justify our answer.
(i) The collection of all months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
(iii) The collection of all boys in your class.
Sol. (i) The collection of all months of a year beginning with the letter J is a well defined collection of objects because one can definitely identify a month that belongs to thisv collection. Hence, this collection is a set.
(ii) The collection of ten most talented writers of India is not a well defined collection because the criteria for determining a writer’s talent may vary from person to person. Hence, this collection is not a set.
(iii) The collection of all boys in your class is a well defined collection because you can definitely identify a boy who belongs to this collection. Hence, this collection is a set.
Methods to Write a Set
(i) Roster Method or Tabular Method :
In this method a set is described by listing elements, separated by commas and enclose then by curly brackets. Note that while writing the set in roster form, an element is not generally repeated e.g. the set of letters of word SCHOOL may be written as {S, C, H, O, L}.
Ex. Write the following sets in roster form:
(i) A = {x: x is an integer and –3 < x <7}.
(ii) B = {x: x is a natural number less than 6}.
Sol. (i) A = {x: x is an integer and –3 <x < 7}
The elements of this set are –2, –1, 0, 1, 2, 3, 4, 5, and 6 only.
Therefore, the given set can be written in roster form as
A = {–2, –1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {x : x is a natural number less than 6}
The elements of this set are 1, 2, 3, 4, and 5 only.
Therefore, the given set can be written in roster form as
B = {1, 2, 3, 4, 5}
(iii) Set builder form (Property Method) : In this we write down a property or rule which gives us all the
element of the set.
A = {x : P(x)} where P(x) is the property by which x A and colon ( : ) stands for ‘such that’
(iv) Symmetric difference of sets :
It is denoted by A △ B and A △ B = (A – B) ⋃ (B – A)
(v) Complement of a set : A' = {x : x A but x U} = U – A
e.g. U = {1, 2,........, 10}, A = {1, 2, 3, 4, 5} then A' = {6, 7, 8, 9, 10}
(vi) Disjoint sets : If A ⋂ B = ф, then A, B are disjoint e.g. If A = {1, 2, 3}, B = {7, 8, 9} thenA ⋂ B = ф
Laws of Algebra of Sets (Properties of Sets)
(i) De-Morgan Laws :
(A ⋃ B)' = A' ⋂ B' ; (A ⋂ B)' = A' ⋃ B'
(ii) A – (B ⋃ C) = (A – B) ⋂ (A – C) ; A – (B ⋂ C) = (A – B) ⋃ (A – C)
(iii) Distributive Laws :
A ⋃ (B ⋂ C) = (A ⋃ B) ⋂ (A ⋃ C); A ⋂ (B ⋃ C) = (A ⋂ B) ⋃ (A ⋂ C)
(iv) Commutative Laws :
A ⋃ B = B ⋃ A ; A ⋂ B = B ⋂ A
(v) Associative Laws :
(A ⋂ B) ⋂ C = A ⋂ (B ⋂ C) ; (A ⋃B) ⋃C = A ⋃(B⋃ C)
(vi) Identity law :
A ⋂ U = A ; A ⋃ ф = A
(vii) Complement law :
A ⋂ A' = U, A ⋃ A' = ф , (A')' = A
(viii) Idempotent law :
A ⋂ A = A, A ⋃ A = A
(ix) A ⋂ ф= ф ; A ⋂ U = A
A ⋃ ф= A ; A ⋃ U = A
(x) A⋂ B ⊆ A ; A ⋂ B ⊆ B
(xi) A⊆ A ⋃ B ; B ⊆ A ⋃ B
(xii) A ⊆ B ⇒ A ⋂ B = A
(xiii) A ⊆ B ⇒ A ⋃ B = A
Ex. Let A = {x : x R, |x| < 1] ; B = [x : x Є R, |x – 1| ≥ 1] and A ⋃ B = R – D, then the set D is-
(A) [x : 1 < x ≤ 2] (B) [x : 1 ≤ x < 2] (C) [x : 1 ≤ x ≤ 2] (D) none of these
Sol. A = [x : x ∊ R, –1 < x < 1]
B = [ x : x R : x – 1 ≤ –1 or x – 1 ≥ 1]
= [x : x R : x ≤ 0 or x ≥ 2]
∴ A ⋃ B = R – D
where D = [x : x ∈ R, 1 ≤ x < 2]
Thus (B) is the correct answer.
Venn Diagram In
NEXT PAGE ≫
TOP video from YOUTUBE
KEYWORDS ::
sets and relations pdf sets and relations formulas
sets and relations class 12 sets and relations symbols
sets and relations all formulas sets and relations questions and answers
sets and relations problems and solutions sets and relations basics