Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 2 Sets will enable students to study smartly.

## CHSE Odisha 11th Class Math Notes Chapter 2 Sets

**Set:**

Set is an undefined term in mathematics. But we understand set as “a collection of well-defined objects”.

- Set is a collection.
- The objects (called elements) in a set must be well-defined.

**Set Notation:**

We denote set as capital alphabets like A, B, C, D…..and the elements by the small alphabets like x, y, z ….

- If x is an element of set A we say “x belongs to A” and write ‘x ∈ A’.
- If x is not an element of set A we say “x does not belong to A” and we write ‘x ∉ A’.

**Set Representation:**

(a) Extension or tabular or Roster Method: In this method, we describe a set listing the elements, separated by commas within curly brackets.

Note: While listing out the elements the repetition of an object has no effect. Thus, we don’t do this.

(b) Intention or set builder or set selector method: In this method: a set is described by a characterizing property p(x) of element x. In this case, the set is described as {x : p(x) holds}

**Types Of Set:**

(a) Empty of full or void set: It is a set with no element.

- We denote empty set by ‘Φ’
- There is only one empty set.

(b) Singleton set: It is a set with only one element.

(c) Finite set: A set is finite if it has a finite number of elements.

(d) Infinite Set: A set that is not finite is called an infinite set.

(e) Equal sets: Two sets A and B are equal if they have the same elements. Two sets A and B are equal if all elements of A are also elements of B and all elements of B are also elements of A.

(f) Equivalent set: Two finite sets A and B are equivalent if they have the same number of elements.

Subsets: Let A and B be two sets. If every element of A is an element of B then A is called a. subset of B (we write A ⊂ B) and B is called a superset of A (We write B ⊃ A)

Thus A ⊂ B is x ∈ A ⇒ x ∈ B

Note.

(i) A set is a subset of itself.

(ii) Empty set Φ is a subset of every set.

(iii) A is called a proper subset of B if B contains at least one element that is not in A

(iv) If A has n elements then total number of subsets of A = 2^{n}.

Universal set:

A set ‘U’ that contains all sets in a given context is called the universal set.

Power set:

Let A is any set. The collection (or set) of all subsets of A is called the power set of A. We denote it as P(A)

P(A) = { S: S ⊂ A }

**Set Operations:**

(a) Union of sets :

The union of two sets A and B is the set of all elements of A or B or both.

∴ A ∪ B = {x ∈ A or x ∈ B}

(b) Intersection of sets:

Intersection of two sets A and B is the set of all those elements that belong to both A and B . (or all common elements of A and B)

∴ A ∩ B = {x: x ∈ A and x ∈ B }

Two sets A and B are disjoint if A ∩ B = Φ. Otherwise, A and B are intersecting or overlapping sets.

(c) Difference of sets: The difference of two sets ‘A and B’ is the set of all elements of A which do not belong to B.

∴ A- B = {x: x ∈ A and x ∈ B)

(d) Symmetric difference of two sets: Symmetric difference of two sets A and B is the set (A – B) ∪ (B – A)

∴ A Δ B = (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B)

(e) Complement of a set: Let the complement of a set A (denoted as A’ or A^{c}) be defined as U – A

- A’ = {x ∈ U) : x ∉ A)
- x ∈ A’ ⇔ x ∉ A

**Laws Of Set Algebra:**

(a) Idempotent law: For any set A we have

(i) A ∪ A = A

(ii) A ∩ A = A

(b) Identity laws: For any set A we have

(i) A ∪ Φ = A and

(ii) A ∩ U = A

(c) Commutative laws: For any three sets A, B, and C

(i) A ∪ B = B ∪ A

(ii) A ∩ B = B ∩ A

(d) Associative laws: For any three sets A, B, and C

(i) A ∪ (B ∪ C) = (A ∪ B) ∪ C

(ii) A ∩ (B ∩ C) = (A ∩ B) ∩ C

(e) Distributive laws: For any three sets A, B, and C

(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

(f) De-morgans laws: For any two sets A and B

(i) (A ∪ B)’ = A’ ∩ B’

(ii) (A ∩ B)’ = A’ ∪ B’

Some more properties of sets: For any three sets A, B, and C

(a) A ⊂ (A ∪ B) and (A ∩ B) ⊂ A

(b) A ∪ B = B ⇔ A ⊂ B

(c) A ∩ B = A ⇔ A ⊂ B

(d) B ⊂ A and C ⊂ A ⇒ (B ∪ C ) ⊂ A and A ⊂ B, A ⊂ C ⇒ A ⊂ (B ∩ C)

(e) B ⊂ C ⇒ A ∪ B ⊂ A ∪ C and A ∩ B ⊂ A ∩ C

(f) A – B = A ∩ B’

(g) A – B = A ⇔ A ∩ B = Φ

(h) (A – B) ∪ B = A ∪ B and (A – B ) ∩ B = Φ

(i) A ⊆ B ⇔ B’ ⊆ A’

(j) A Δ B = B Δ A

Cardinality or order of a finite set: The cardinality or order of a finite set A (denoted as |A| or O(A) or n (A)) is the number of elements in ‘A’.

Some important results on the cardinality of finite sets and applications of set theory:

If A, B, and C are finite sets and ‘U’ is the finite universal set then a number of elements belonging to at least one of A or B.

(a) |A ∪ B| = |A| + |B| – |A ∩ B|

(b) |A ∪ B| = |A| + |B| for A ∩ B = Φ i.e. for two disjoint sets A and B

(c) Number of elements belonging to at least one of A, B, or C

= |A ∪ B ∪ C|

= |A| + |B| + |C| – |A ∩ B| – |B ∩ C| – |C ∩ A| + |A ∩ B ∩ C|

(d) Number of elements belonging to exactly two of the three sets A, B, and C = |A ∩ B| + |B ∩ C| + |C ∩ A| – 3 |A ∩ B ∩ C|

(e) Number of elements belonging to exactly one of the three sets A, B, and C = |A| + |B| + |C| – 2 |A ∩ B| -2 |B ∩ C| – 2 |C ∩ A| + 3 |A ∩ B ∩ C|

(f) Number of elements belonging to A but not B = |A – B| = |A| – |A ∩ B|

∴ |A| = |A – B| + |A ∩ B|

(g) Number of elements belonging to exactly one of A or B

= |A Δ B| = |A – B| + |B – A|

= |A| + |B| – 2 |A ∩ B|

(h) |A’ ∪ B’| = |U| – |A ∩ B|

(i) |A’ ∩ B’| = |U| – |A ∪ B| = Number of elements belonging to neither A nor B.