Mathematics Definitions , Key Concepts , points to remember & formulas of sets

DEFINITIONS

\color{green}( ✍️ "Set")
A color{blue} ul(mathtt ("Set")) is a WELL-Defined collection of objects. ( called elements of sets )

\color{green} {(i)" 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 { }.

\color{green} {"(ii) 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.

\color{green}( ✍️ "Empty set")
A 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 \color{blue}(Phi) or \color{blue}({ }).

\color{green}( ✍️ "Finite and infinite set")
A set which is empty or consists of a definite number of elements is called "finite" otherwise, the set is called "infinite."

\color{green}( ✍️ "Cardinality")
"Cardinality" of a finite Set is defined as total number of elements in a set.

\color{green}( ✍️ "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.

\color{green}( ✍️ "Subset")
A set A is said to be a subset of a set B if every element of A is also an element of B.

\color{green} (✍️ "Proper subset and Superset")
Let A and B be two sets. If A ⊂ B and A ≠ B , then A is called a proper subset of B and B is called superset of A.

\color{green}( ✍️ "Singleton set: ")
If a set A has only one element, we call it a singleton set. Thus ,{ a } is a singleton set.

\color{green}( ✍️ "Power Set")
The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.

\color{green}( ✍️ "Universal Set")
Universal Set is the set defines as the set containing all objects or elements and of which all other sets are subsets.

 \color{green}( ✍️ "Venn Diagrams ")
The universal set is represented usually by a rectangle and its subsets by circles or ellipses.

\color{green}( ✍️ "Union of sets ")
The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both).
we write A ∪ B and usually read as "‘A union B’."
The common elements being taken only once.
A ∪ B = { x : x ∈ A or x ∈B }

\color{green}( ✍️ "Intersection of sets : ")
The intersection of two sets A and B is the set of all those elements which belong to both A and B.
Symbolically, we write A ∩ B = {x : x ∈ A and x ∈ B}.

\color{green}( ✍️ "Disjoint sets")
If A and B are two sets such that A ∩ B = Phi , then A and B are called disjoint sets.

\color{green}( ✍️ "Difference of sets (A-B) or (A/B) : ")
The difference of the sets A and B in this order is the set of elements which belong to A but not to B.
Symbolically, we write A – B and read as "“ A minus B”"

\color{green}( ✍️ "Complement of a Set ")
Let U be the universal set and A a subset of U. Then the complement of A is the set of all elements of U which are not the elements of A.
we write A′ to denote the complement of A with respect to U.

KEY CONCEPTS

\color{green} ✍️ A set is well-defined if
There is no ambiguity as to whether or not an object belongs to it, i.e we can always tell what is and what is not a member of the set.

\color{green} ✍️ If color{green} (a \ \ "is an element of a set A"), we say that "“ a belongs to A”." Thus, we write a ∈ A.

\color{green} ✍️ (i) If same element comes twice or more time in set, then it is consider only once at final set.

(ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.

(iii) The elements of a set are represented by small letters a, b, c, x, y, z, etc.

\color{green} ✍️ It may be noted that while writing the set in roster form an element is not generally repeated, i.e., all the elements are taken as distinct.

\color{green} ✍️ It is not possible to write all the elements of an infinite set within braces { } because the numbers of elements of such a set is not finite.

So, we represent some infinite set in the roster form by writing a few elements which clearly indicate the structure of the set followed ( or preceded ) by three dots.

\color{green} ✍️ A set does not change if one or more elements of the set are repeated.

\color{green} ✍️ Every set A is a subset of itself, i.e., A subset A.

\color{green} ✍️ Since the empty set φ has no elements, we agree to say that φ is a subset of every set.

\color{green} ✍️ (a,b] is also written as  ]a,b]

\color{green}✍️ The number (b – a) is called the length of any of the intervals (a, b), [a, b], [a, b) or (a, b].

\color{green}✍️ The sets A – B, A ∩ B and B – A are mutually disjoint sets, i.e., the intersection of any of these two sets is the null set

\color{green}✍️ If A is a subset of the universal set U, then its complement A′ is also a subset of U.

points to remember

if A is a set with n(A) = m, then it can be shown that n [ P(A)] = 2^m

\color{green} ✍️ A-B also represented as A//B

\color{green}( ✍️ "Some Properties of the Operation of Union : ")
(i) A ∪ B = B ∪ A (Commutative law)

(ii) ( A ∪ B ) ∪ C = A ∪ ( B ∪ C) (Associative law )

(iii)  A ∪ φ = A (Law of identity element, φ is the identity of ∪)

(iv) A ∪ A = A (Idempotent law)

\color{green}( ✍️ "Some Properties of Operation of Union Intersection : ")
(i) A ∩ B = B ∩ A (Commutative law).

(ii) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law).

(iii)  φ ∩ A = φ, U ∩ A = A (Law of φ and U).

(iv) A ∩ A = A (Idempotent law)

(v) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law ) i. e., ∩ distributes over ∪(Proof is shown in Fig. )

(v) A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C )

\color{green}( ✍️ "Some Properties of Complement Sets ")
(i). A ∪ A′ = U

(ii) A ∩ A′ = φ (Complement laws)

(iii) (A ∪ B)´ = A′ ∩ B′

(iv) (A ∩ B )′ = A′ ∪ B′ (De Morgan’s law)

(v) (A′ )′ = A (Law of double complementation )

(vi). φ′ = U and U′ = φ (Laws of empty set and universal set)

formulas

\color{green}( ✍️ "Important Formula on operation of Sets ")
1. A-B = A∩B'

2. A-(B∩C) = (A-B) ∪ (A-C)

3. A-(B∪C) = (A-B) ∩ (A-C)

4.A∩(B-C) = (A∩B) - (A∩C)

\color{green}( ✍️ "Formula of union and Intersection : ")
If A and B are finite sets, then
 n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B )

- If A ∩ B = φ, then

 n ( A ∪ B ) = n ( A ) + n ( B )

 n ( A ∪ B) = n ( A – B) + n ( A ∩ B ) + n ( B – A )

- If A, B and C are finite sets, then

n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C) – n ( A ∩ C ) + n ( A ∩ B ∩ C )

n [ A ∩ ( B ∪ C ) ] = n ( A ∩ B ) + n ( A ∩ C ) – n [ ( A ∩ B ) ∩ (A ∩ C)]