1. Sets

1.1

1.2

1.3

Types of sets

Null set

• Φ (phi) or {  }
• Properties of Empty Set

Property 1: Empty Set is a Subset of Every Set.

According to the property, the empty or null can be regarded as a subset of any set. That is, given a set P, the empty set is a subset of P, such that Φ ⊆ P; ∀ P. Let's look at an example to better understand the attribute.

Property 2: Union with Empty Set

The property asserts that the union of any set with an empty set always results in the set itself. Q ⋃ Φ = Q; ∀ Q. is a mathematical expression.

Property 3: Intersection with Empty Set

This property, in contrast to the union property, states that the intersection of any given set's sets with an empty one will always result in an empty set. E ⋂ Φ = Φ; ∀ E is the mathematical expression.

Property 4: Cardinality of Empty Set

Cardinality specifies the total number of items in a given set. As we know, an empty kind of set has no items and so has 0 cardinality. This is represented as |Φ| = 0.

Property 5: Power Set of Empty Set

An empty set has no items. As a result, the power set of an empty set{} may be defined as P(Φ) = {Φ}.

An empty or null set's power set contains exactly:

• A set that includes a null set.
• It contains zero or null items.
• The sole subset is the empty set.
• As a result, the number of power set components is 20 = 1
• As a result, the power set only has one element, which is the empty set itself.

Property 6: Cartesian Product of Empty Set

The multiplication of the provided sets is defined by the Cartesian product. Taking the cartesian product of any set with an empty/null set always yields an empty result. The explanation is obvious: the empty set includes no items. Mathematically expressed as: B x Φ = Φ; ∀ B

Property 7: Complement of Empty Set

The universal set is the complement of a set that is an empty/null set. The ultimate solution is a universal set itself if we carry the difference between a universal set and an empty type set. 'Φ’ = U is the mathematical notation. This is because the universal set contains all items, but the empty set has none.

Singleton set

A set is referred to as a singleton set when the number of elements in the set is one. The set of the form S = {a} is called a singleton set. The singleton sets are always finite as it has only 1 element. The symbol of the singleton set is written as: P = {x}

Cardinality of Singleton Set

The cardinality of singleton set is 1 as the number of elements present in the singleton set is 1. Since the singleton set contains only one element the cardinal number of singleton set is 1.

Power Set of a Singleton Set

The power set of a singleton set contains two elements the null set and the singleton set itself. The cardinality of the power set of a singleton set is 2 as it contains only two elements.

Is Zero Set {0} a Singleton Set?

Zero set is a set with one element zero. The cardinality of zero set is 1. So, the zero set i.e., {0} is a singleton set. Zero set and null set are different as zero set represents a set with one element zero whereas the null set is a set with no elements. Since a set with one element is singleton set so, the zero set is also a singleton set. Zero set S = {0} is a singleton set

Infinite set

Finite set

Universal Set

A universal set is a set that contains all the elements of the rest of the sets. It can be said that all the sets are the subsets of Universal sets. The universal set is denoted as U.

Properties of Universal Set:

• Contains all the elements being discussed.
• Every set is a subset of the universal set.
• The complement of a set is taken disjointed from the universal set.
• It is usually finite or defined by context (can be infinite in some cases).
• The intersection of U with any set A is A:
  • U ∩ A = A
• The union of U with any set A is U:
  • U ∪ A = U

Subset

• Set A will be called the Subset of Set B if all the elements present in Set A already belong to Set B.
• The symbol used for the subset is ⊆.
• If A is a Subset of B, it will be written as A ⊆ B.
• • Key Rule: It is possible for $A$ to be exactly the same as $B$
• Analogy: Think of "$\leq$" (less than or equal to) in algebra.
• The number of subsets of a set with n elements is 2n. This includes both proper and improper subsets.

Properties of Subsets :

• A ⊆ A (Every set is a subset of itself)
• ∅ ⊆ A (Empty set is a subset of every set)
• If A ⊆ B and B ⊆ C, then A ⊆ C (transitive property)
• If A ⊆ B and A ≠ B, then A is a proper subset of B (written A ⊂ B)
• A set with n elements has 2ⁿ subsets (including ∅ the set itself)

Proper subset

• A proper subset is a subset that contains some but not all elements of the original set. It is not equal to the original set. If A and B are sets, and A is a proper subset of B, we write: A ⊂ B.
• Analogy: Think of "<" (strictly less than) in algebra.
• The number of proper subsets of a set with n elements is 2n - 1.
• Set $B$ must contain at least one extra element that is not in $A$
  

Example

Let B = {1, 2, 3} Proper subsets of B are: ∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} {1, 2, 3} is a subset of B, but not a proper subset, because it is equal to B.

Properties of Proper Subset:

• A proper subset has fewer elements than the original set.
• Every proper subset is a subset, but not every subset is proper.
• The empty set (∅) is a proper subset of any non-empty set.
• The original set is not a proper subset of itself.

Improper subset

What will be the improper subset of set A = {1, 3, 5}? Improper subset: {1,3,5} Number of improper subsets: For any set with n elements, there is exactly one improper subset — the set itself.

• The number of improper subsets of any set is always 1.
  

Difference between proper and improper subset

Difference between Subset and Superset

Equal set

Two sets having the same elements and an equal number of elements are called equal sets. The elements in the set may be rearranged, or they may be repeated, but they will still be equal sets.

Properties of Equal Sets :

• Two sets are equal if every element of one set is also in the other set.
• The order of the elements doesn't matter.
• The repetition of elements doesn’t matter (sets don’t allow duplicates).
• If A = B, then A ⊆ B and B ⊆ A.
• Equal sets have the same number of elements (same cardinality).

Equivalent set

Equivalent Sets are those which have the same number of elements present in them. It is important to note that the elements may be different in both sets but the number of elements present is equal.

Properties of Equivalent Sets :

• Equivalent sets have the same number of elements.
• The elements can be different; it only the count matters.
• If two sets A and B are equivalent, we write: A ≈ B
• All equal sets are equivalent, but not all equivalent sets are equal.

Disjoint set

That have two sets A and B that do have no common elements are called Disjoint Sets. The intersection of the Disjoint set is ϕ, now for set A and set B A∩B = ϕ.

Properties of Disjoint Sets

• No common elements → A ∩ B = ∅
• If sets are disjoint, their intersection is the empty set.
• Disjoint sets can be finite or infinite.
• Two sets can be disjointed even if one or both are empty.
• In Venn diagrams, disjoint sets do not overlap.

Power set

The set of any set A is defined as the set containing all the subsets of set A. It is denoted by the symbol P(A) and read as the Power set of A.

For any set A containing n elements, the total number of subsets formed is 2n. Thus, the power set of A, P(A) has 2n elements.

It includes:

• The empty set (∅)
• All individual elements as sets
• All combinations of elements
• The set itself

Properties of Power Set :

1. If a set has n elements, its power set has 2ⁿ subsets. Example: A = {a, b, c} → 3 elements → power set has 2³ = 8 subsets
2. The power set always includes: The empty set (∅) The original set
3. Power set of ∅ is: P(∅) = {∅}
4. The power set is always larger than the original set (except when the set is empty).

Example:

For any set A = {a,b,c}, the power set of A is? Solution: Power Set P(A) is, P(A) = {ϕ, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}}

Intervals

Union of intervals

The union of two intervals P and Q contains all the elements present in one or both the intervals. Examples Some examples of union of two intervals are given below.

• (2, 7) ∪ (4, 9) = (2, 9)
• [1, 10] ∪ [5, 15] = [1, 15]
• (5, 12) ∪ [6, 14] = (5, 14]