1. Sets - Class 11 Mathematics English CBSE Notes
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1. Sets - Class 11 Mathematics English CBSE Notes
1. Sets
SETS
Set: A collection of well defined objects is called a set.
- Objects of collection is called "element" or "member".
Properties:
(i) All elements of a set should have special property.
(ii) They all should be differ to each other or they do not repeat.
(iii) They should be well defined.
- Name of any set is writen in capital letter of English Alphabets
A, B, C, D ........................ X, Y, Z
- Elements and members of a set is writen in small letter of English Alphabets or using number system within a brace (medium bracket) with using commas { }
Example : A = {a, b, c, d} Or B = {1, 2, 3, 4}
Standard symbols of some special sets:
N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
R : the set of real numbers
Z+ : the set of positive integers
Q+ : the set of positive rational numbers, and
R+ : the set of positive real numbers.
Symbols and their meaning:
(i) " ∈ " (epsilon) : "belong to"
If a is an element of a set A, we say that “ a belongs to A”
in symbolic form we write it as : a ∈ A
(ii) " ∉ " : "not belong to"
When any element which not belong to any given set then we use symbol ∉ : "not belong to"
if a is not an element of set A. we say that "a not belong to A"
in symbolic form we write it as : a ∉ A
(iii) " ⇒ " : emplies
In common language "⇒" means "emply toward ..... also ......"
(iv) " = " : Equal to
A = B, it means set A is equal to set B.
(v) " ≠ " : Not equal to
A ≠ B, It means set A not equal to set B.
(vi) “⇔” is a symbol for two way implications, and is usually read as " if and only if ".
(vii) " : " : colon Here in the set colon stands for "such that"
Representation of sets:
Every set is introduced by its elements. So for expressing a set before we have to express its elements.
Now, Method to express the elements of a set.
(1) Roster Or tabular form :
In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }. For example,
(i) First five natural numbers:
A = {1, 2, 3, 4, 5}.
(ii) First four letter of English Alphabets
B = {a, b, c, d}.
(iii) The set of all natural numbers which divide 42
C = {1, 2, 3, 6, 7, 14, 21, 42}.
(iv) The set of all vowels in the English alphabet.
D = {a, e, i, o, u}.
Note:
In Roster Form no-element is generally repeated
Example:
The set of letters forming the word 'ELEMENT'
E = {E, L, M, N, T}
Here no-element has been repeated.
(2) Set-builder form
In set-bulder form, all the elements of a set possess a single common property
which is not possessed by any element outside the set.
Example:
D = { a, e, i, o, u}
All elements of set A have a common property e.i vowel in English alphabet.
Then, in set-builder form:
D = { x : x is a vowel in english alphabet }
This set will be read as :
" the set of all x such that x is a vowel in english alphabet. "
C = {y : y is a natural number which divides 42}
B = (z : z is a first four letter of english alphabet}
Other examples
(i) A = {1, 2, 9, 25} Roster form
Set-builder form
A = {x : x = n2, where n ∈ N and n < 6}
(ii) B = {P, R, I, N, C, A, L}
Set-builder form
B = {x : x is a letter of the word PRINCIPAL}
(iii) C = {1, 2, 3, 6, 9, 18}
Set-builder form
C = { x : x is a positive integer and is a divisor of 18}
(iv) D = (3, 6, 9, 12}
Set-builder form
D = {x : x = 3n where n ∈ N and 0 < n < 5}
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