### NULL/ VOID/ EMPTY SET

A set which has no element is called the null set or empty set andis denoted by ϕ ≤ phi). The number of elements of a set A is denoted as n ≤ A) and n ≤ ϕ)* = *0 as it contains no element. For example the set of all real numbers whose square is –1.

### SINGLETON SET

A set containing only one element is called Singleton Set.

### FINITEANDINFINITE SET

A set, which has finite numbers of elements, is called a finite set. Otherwise it is called an in finite set. For example, the set of all days in a week is a finite set whereas; the set of all integers is an infinite set.

### UNIONOF SETS

Unionof two or more sets is the set of all elements that belong to any of these sets. The symbol used for union of sets is ‘∪’ i.e.A∪B* = *Union of set A and set B* = *{x: x ∈ A or x∈B ≤ or both)}

**Example:** A* = *{1, 2, 3, 4} and B* = *{2, 4, 5, 6} and C* = *{1, 2, 6, 8}, then A∪B∪C* = *{1, 2, 3, 4, 5, 6, 8}

### INTERSECTION OF SETS

It is the set of all the elements, which are common to all the sets. The symbol used for intersection of sets is ‘∩’ i.e. A ∩ B* = *{x: x ∈ A and x∈ B}

**Example:**If A* = *{1, 2, 3, 4} and B* = *{2, 4, 5, 6} and C* = *{1, 2, 6, 8}, then A ∩ B ∩ C* = *{2}

### DIFFERENCE OF SETS

The difference of set A to B denoted as A – B is the set of those elements that are in the set A but not in the set B i.e. A – B* = *{x: x∈ A and x ∉ B}

Similarly B – A *=* {x: x∈B and x∉ A}

In general A-B ≠ B-A

** Example: **If A* = *{a, b, c, d} and B *= *{b, c, e, f} then A-B* = *{a, d} and B-A* = *{e, f}.** **

**Symmetric Difference of Two Sets:**

For two sets A and B, symmetric difference of A and B is given by ≤ A – B) ∪ ≤ B – A) and is denoted by A Δ B.

### SUBSET OF A SET

A set A is said to be a subset of the set B if each element of the set A is also the element of the set B. The symbol used is ‘⊆’ i.e.A ⊆ B ↔ ≤ x ∈A⇒ x ∈ B).

Each set is a subset of its own set. Also a void set is a subset of any set. If there is at least one element in B which does not belong to the set A, then A is a proper subset of set B and is denoted as A ⊂ B. e.gIf A* = *{a, b, c, d} and B* = *{b, c, d}. Then B⊂A or equivalently A⊃B ≤ i.e A is a super set of B). Total number of subsets of a finite set containing n elements is 2^{n}.

**Equality of Two Sets:**

Sets A and B are said to be equal if A⊆B and B⊆A; we write A = B

**Some More Results Regarding the Order of Finite Sets:**

Let A, B and C be finite sets and U be the finite universal set, then

i). n ≤ A ∪ B) = n ≤ A) + n ≤ B) – n ≤ A ∩ B)

ii). If A and B are disjoint, then n ≤ A ∪ B) = n ≤ A) + n ≤ B)

iii). n ≤ A –B) = n ≤ A) – n ≤ A ∩ B) i.e. n ≤ A) = n ≤ A – B) + n ≤ A ∩ B)

iv). n ≤ A ∪ B ∪ C) = n ≤ A) + n ≤ B) + n ≤ C) – n ≤ A ∩ B) – n ≤ B ∩ C) – n ≤ A ∩ C) + n ≤ A ∩ B ∩ C)

v). n ≤ set of elements which are in exactly two of the sets A, B, C)

= n ≤ A ∩B)+n ≤ B ∩ C) + n ≤ C ∩ A) –3n≤ A ∩ B ∩ C)

vi). n≤ set of elements which are in atleast two of the sets A, B, C)

= n ≤ A ∩ B) + n ≤ A ∩ C) + n ≤ B ∩ C) –2n≤ A ∩ B ∩ C)

vii). n ≤ set of elements which are in exactly one of the sets A, B, C)

= n ≤ A) + n ≤ B) + n ≤ C) – 2n ≤ A ∩ B) – 2n ≤ B ∩ C) – 2n ≤ A ∩ C) + 3n ≤ A ∩ B ∩ C)** **

### DISJOINT SETS

If two sets A and B have no common elements i.e. if no element of A is in B and no element of B is in A, then A and B are said to be Disjoint Sets. Hence for Disjoint Sets A and B n ≤ A ∩ B)* = *0.

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CBSE Class 11 Maths Sets Relations and Functions All Topic Notes CBSE Class 11 Maths All Chapters Notes