CLASS 11 Sets (Basics of Mathematics) Notes

CLASS 11 Sets (Basics of Mathematics) Notes

  March 15, 2023    Schoolix

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SETS

# Definition 

Collection of well defined objects.

    e.g - List of all students studying in class 11th

# Representation of sets

A) Roaster/Tabular Form:- All the elements are listed, separated by commas and in closing curly                                                        brackets.

Ex: Natural number less than 5

A={1,2,3,4}

B) Set Builder Form:- A statement is written here in in this form such that all the element of set have a                                        single common property which is not possessed by any element outside the set.

A={x: x ∈ N, x<5}

# Type of Sets

A) Null/Empty/Void Set:- Contains no element.

Ex:- A= { }

B) Single ton Set:- Contains only one element.

Ex:- A={1}

C) Finite Set:- Finite number of element.

Ex:- A={1,2,3,4,5}

D) Infinite Set:- Contains Infinite number of element.

Ex:- A={x: x ∈ N}

E) Universal Set:- Superset of all set.

Ex:- U={x ∈ R}

F) Equal Set:- Two sets having same element.

G) Equivalent Set:- Sets having same cardinal number.

👉 Cardinal Number/Order of finite set:- Number of distinct element in the set.

Notations:- n(A),o(A),C(A),|A|

# Subsets

A set (A) is said to be subset of set (B), if every element of set (A) is also an element of set (B).

Ex:- A={1,2}        B={1,2,3,5}

A⊂B

NOTE:- N⊂W⊂Z⊂Q⊂R  

👉 ϕ is always an subset of any set.

👉 Number of subset of any set = 2ⁿ, where n=n(A). 

^{👉Number of proper subsets (All subsets excluding complete set) =} 2ⁿ-1

# Power Set

Defined as set of subsets of any set.

Ex:- A={1,3}

Then, P(A)={ϕ,{1},{3},{1,3}}

NOTE:- If n(A)= n

              Then, n(P(A))=2ⁿ

              N(P(P(A)))= 2**2**2

# Operations on Set

A) Union of Sets(U):- Present in either 'A' or 'B'.

Ex:- A={1,2}     B={2,3}

AUB={1,2,3}

👉 Properties

a) AUB=BUA                                                         d) AUU = U

b) (AUB)UC = AU(BUC)                                      e) AUϕ = A

c) AUA = A

B) Intersection of Sets (∩):- Present in both.

Ex:- A={1,2}     B={2,3}

A∩B= {2}

👉 Properties

a) A∩B=B∩A                                                       e) A∩U = A

b) A∩(B∩C) = (A∩B)∩C                                    f) A∩(BUC) = (A∩B)U(A∩C)

c) A∩A = A                                                          g) AU(B∩C) = (AUB)∩(AUC)

d) A∩ϕ = ϕ

C) Complement of Set (A^c):- All except 'A' part.

A^c = U-A

👉 Properties

a) A^c = U-A                                                        

b) Complement A^c of A^c = A

Now According to De-Morgan's Theorem

c) (A ∩B)^c = A^cUB^c

^d) (AUB)^c = A^c∩B^c

^{D) Difference of Two Sets:- The difference of set A and B in this order is the set of element which                                                         belongs to A but not B.}

Ex:- A={1,2}     B={2,3}

Then, A-B= {1}

👉 Properties

a) A-A = ϕ                                                           d) U-ϕ = U

b) A-ϕ = A                                                           e) A-U = ϕ

c) U-U = ϕ

NOTE:- a) If A∩B = ϕ, then A,B Sets are called DISJOINT SETS.

D) Symmetric difference of two sets:- For two sets A and B, symmetric difference of these sets is the                                                                    set consisting of elements of set A and B taken together except                                                                    common element.

Ex:- A={1,2}     B={2,3}

A△B = {1,3}

# Important formula 

A) n(AUB) = n(A) + n(B) - n(A∩B)

B) n(AUBUC) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(A∩C) + n(AUBUC)