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


`\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.`


`\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)


`\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)]`