Class 11 Set Chapter notes||Class 11 Set notes||CBSE Class 11 maths chapter 1 notes||VECTOR STUDIES
SET THEORY
- It is given by GEORGE CANTOR.
- It is used to define the concept of relation and function,Geometry,sequences and probability.
- A set is a well-defined collection of distinct objects.
- e.g The collection of all vowel in alphabet forms a set
There are two methods of representing a set;
- 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 { }
E.g. the set of all number in a dice is described in roster form as {1,2,3,4,5,6}.
Points to be noted in roster form:
- In roster form, the order in which the elements are listed is immaterial. g. The set of all vowels in the English alphabet can be written as {a, e, i, o, u} or {a,u,i,o,e} or {u, e, i, o, a} or {o, e, i, a, u}
- The dots at the end tell us that the list of odd numbers continue indefinitely. E.g.: The set of odd natural numbers is represented by {1, 3, 5, . . .}.
- In roster form, an element is not generally repeated, i.e., all the elements are taken as distinct.
E.g. The set of letters forming the word ‘SCHOOL’ is { S, C, H, O, L} .
- Set Builder Form-
In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
- In the set {a, e, i, o, u}, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property.
Denoting this set by V, we write V = {x : x is a vowel in English alphabet}.
- Please note that any other symbol like the letters y, z, etc. could be used.
- The symbol should be followed by a colon “ : ”.
- After the sign of colon, we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces.
- If a set of number doesn’t follow any pattern, it can’t be written in set builder form.
- The objects in a set are known as - MEMBER/ELEMENTS.e.g-{a,e,i,o,u}.In this set of vowels a,e,i,o,u are elements or menbers of the set.
- The set are denoted with the capital letter of the english alphabet.e.g- A={1,2,3}.
- If a is an element of a set A, we say that “ a belongs to A” the Greek symbol ∈ (epsilon) is used to denote the phrase ‘belongs to’. Thus, we write a ∈
- If ‘b’ is not an element of a set A, we write b ∉ A and read “b does not belong to A”.
- P is set of prime factors of 30, then 3 ∈ P but 15 ∉ P.
- FINITE SET-It is a type of set in which elements are finite..eg-A={1,2,3,4}.
- INFINITE SET-It is a type of set in which elements are infinite.e.g-B={1,2,3,........}.
EMPTY OR NULL OR VOID SET
- Set which does not contain any element is called the empty set or the null set or the void set. The empty set is denoted by the symbol φ or { }.
- Examples of empty sets. Let A = {x : 5 < x < 6, x is a natural number}. Then A is the empty set, Thus we dente A set by the symbol φ or { }.
EQUAL SET
- Two sets A and B are said to be equal if they have exactly the same elements and we write A = B.
Otherwise, the sets are said to be unequal and we write A ≠ B.
- Examples :
Let A = {1, 2, 3, 4} and B = {3, 1, 4, 2}. Then A = B.
Let C = {1, 2, 3, 4} and D = {1,2,3,5}. Then C ≠ D.
SINGLETON SET
- If a set A has only one element, Than it is called a singleton set. Thus { a } is a singleton set.
- E.g. C={x : x ∈ N+ and x2 = 4} , it has only one element C={2}
- If the cardinal no of two sets are same then it is called EQUIVALENT SET.
SUB SET
- A set A is said to be a subset of a set B if every element of A is also an element of B.
- Consider set A = set of all students in your class, B = set of all students in your School.
We note that every element of A is also an element of B; we say that A is a subset of B.
- A is subset of B is expressed in symbols as A ⊂ B. The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’.
- It follows from the above definition that every set A is a subset of itself, i.e. A ⊂ A.
- Since the empty set φ has no elements, we agree to say that φ is a subset of every set.
SUPER SET
Let A and B be two sets. If A ⊂ B and A ≠ B , B is called superset of A.
- The set Q of rational numbers is a subset of the set R of real numbers. We write Q ⊂ R
- Let A = {1, 3, 5} and B = {x : x is an odd natural number less than 6}. Then A ⊂ B and B ⊂ A and hence A = B.
- Let A = { a, e, i, o, u} and B = { a, b, c, d}. Then A is not a subset of B, also B is not a subset of A.
- Let A={1,2,3,4} and B={1,2,3,4,5,6}, then A is subset of B, and B is super set of A.
- Some relation in well defined sets: N ⊂ Z ⊂ Q ⊂ R
N :{ 1, 2, 3, 4, 5…} Natural number
Z: (-7, -6, -5, 1, 4, 5, ..} Integers
Q: { 1.2, 1.3, 1.5, 2.2 ..} Rational Numbers
R: {pie, 1, 3..} Real number
POWER SET
- The collection of all subsets of a set A is called the power set of A.
E.g. Consider the set {1, 2}. Let us write down all the subsets of the set {1, 2}.
Subsets of {1,2} are: φ, { 1 }, { 2 } and { 1, 2 }.
The set of all these subsets is called the power set of { 1, 2 }.
UNIVERSAL SET
- A set that contains all sets in a given context is called the universal set. It is denoted by U.
VENN-DIAGRAMS
- Venn diagrams are the diagrams, which represent the relationship between sets.
- In Venn-diagrams the universal set U is represented by point within a rectangle and its subsets are represented by points in closed curves (usually circles) within the rectangle.
- Union of sets: The union of two sets A and B, denoted by A ∪ B is the set of all those elements which are either in A or in B or in both A and B. Thus, A ∪ B = {x : x ∈ A or x ∈ B}.
- Intersection of sets: The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements which are common to both A and B.Thus, A ∩ B = {x : x ∈ A and x ∈ B}
- Disjoint sets: Two sets A and B are said to be disjoint, if A ∩ B = Φ.
- Difference of sets: For any sets A and B, their difference (A – B) is defined as a set of elements, which belong to A but not to B.Thus, A – B = {x : x ∈ A and x ∉ B},also, B – A = {x : x ∈ B and x ∉ A}
- Complement of a set: Let U be the universal set and A is a subset of U. Then, the complement of A is the set of all elements of U which are not the element of A.Thus, A’ = U – A = {x : x ∈ U and x ∉ A}
SOME PROPERTIES ON OPERATION OF SETS
- A ∪ A’ = ∪
- A ∩ A’ = Φ
- ∪’ = Φ
- Φ’ = ∪
- (A’)’ = A
Idempotent Laws: For any set A, we have
- A ∪ A = A
- A ∩ A = A
Identity Laws: For any set A, we have
- A ∪ Φ = A
- A ∩ U = A
Commutative Laws: For any two sets A and B, we have
- A ∪ B = B ∪ A
- A ∩ B = B ∩ A
Associative Laws: For any three sets A, B and C, we have
- A ∪ (B ∪ C) = (A ∪ B) ∪ C
- A ∩ (B ∩ C) = (A ∩ B) ∩ C
Distributive Laws: If A, B and Care three sets, then
- A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
De-Morgan’s Laws: If A and B are two sets, then
- (A ∪ B)’ = A’ ∩ B’
- (A ∩ B)’ = A’ ∪ B’
