CBSE Class 11 Maths Notes: Sets – Types of Set

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 2ⁿ.

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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