Disjoint Sets | Complement of a set | Difference between two sets | Symmetric difference between two sets

Disjoint Sets

Disjoint sets are sets that have no elements in common. In other words, if two sets are disjoint, their intersection is an empty set.

Formally, sets A and B are disjoint if and only if their intersection is the empty set:

A ∩ B = ∅

If A and B are disjoint sets then n(A ∩ B) = |A ∩ B| = 0

So, from Addition theorem 

∴ |A⋃B| = |A| + |B| - |A ∩ B|

∴ |A⋃B| = |A| + |B| - 0

∴ |A⋃B| = |A| + |B|                         (When A and B are disjoint sets)

In simple terms, if two sets do not share any elements, they are considered disjoint.It's vice-versa is also true. 

Here are a few key points about disjoint sets:

1. Empty Intersection: The defining property of disjoint sets is that their intersection is empty. This means that there are no elements that belong to both sets simultaneously.

2. Example: Let's consider two sets A = {1, 2, 3} and B = {4, 5, 6}. Since there are no common elements between A and B, their intersection is empty: A ∩ B = ∅. Therefore, A and B are disjoint sets.

3. Multiple Sets: Disjoint sets can also be extended to more than two sets. If three or more sets have empty intersections with one another, they are considered pairwise disjoint or mutually disjoint.

Three sets A, B, C are said to be disjoint if they are pairwise disjoint like: A ∩ B = ∅ , B ∩ C = ∅ , C ∩ A = ∅

Consider 'n' sets A₁, A₂, A₃, A₄, A₅, A₆, A₇,.............,A_n-1, A_n are said to be disjoint sets if they are pairwise disjoint. For every  A_i ∩ A_j= ∅  (where 1≤ i ≤ n , 1 ≤ j ≤ n and  i ≠ j )

4. Disjoint Union: The concept of disjoint sets is closely related to the concept of a disjoint union. When combining disjoint sets, the resulting set is a union of the individual sets, where no elements are repeated.

Disjoint sets are often used in various areas of mathematics, such as set theory, probability theory, and graph theory. They allow for clear separation and analysis of different elements or subsets within a larger set.

Complement of a set

The complement of a set is a concept in set theory that refers to the elements that are not contained in a given set, with respect to some larger set called the universal set.

Formally, if A is a subset of the universal set U, then the complement of A, denoted by A' or A^c, or U-A or ~A  or not A  is defined as the set of all elements in U that are not in A.

The complement of A is represented as:

A' = {x ∈ U : x ∉ A}

In other words, A' consists of all elements that belong to U but do not belong to A.

For Example: Suppose we have a universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and a subset A = {2, 4, 6, 8, 10}

then,

∴ A' = U - A

∴ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {2, 4, 6, 8, 10}

∴ A' = {1,3,5,7,9}

Some Important things:

• A' = U - A
• A ⋃ A' = U
• (A')' = A
• A ∩ A' = ∅  So, We can say that Set A and A' are disjoint sets.
• ∴ n(A') = n(U) - n(A)
• U' = ∅
• ∅' = U

The concept of complement is fundamental in set theory and is used in various applications, such as logic, probability, and set operations. It allows for the analysis of elements that do not belong to a specific set and helps define relationships between sets within a universal set.

Difference between two sets 

The difference between two sets is an operation in set theory that represents the elements that are present in one set but not in the other. It is denoted by the symbol "-".

Given two sets A and B, the difference of A and B, denoted as A - B is defined as the set of elements that are in A but not in B.

Mathematically, the difference of A and B is represented as:

A - B = {x : x ∈ A and x ∉ B}

In other words, it includes all elements from set A that are not present in set B.

Some Important things:

• A - B  :  A difference B
• B - A  :  B difference A
• A - B  ≠  B - A
• A - B = A - (A ∩ B)  =  (A ⋃ B) - B  =  A ∩ B'
• B - A = B - (A ∩ B)  =  (A ⋃ B) - A  =  B ∩ A'
• n(A - B) = n(A) - n(A ∩ B)
• n(B - A) = n(B) - n(A ∩ B)

Here are some key points about the difference between two sets:

1. Elements Only in A: The difference of A and B includes all elements that are exclusively present in set A. It does not contain any elements that are common to both sets.

2. Empty Set: If all elements of set A are also present in set B, then the difference of A and B will be the empty set (∅). This happens when set A is a subset of set B.

3. Non-Commutative: The difference operation is not commutative, which means that A - B is not necessarily the same as B - A. The result depends on the order of the sets. (A - B  ≠  B - A)

4. Venn Diagram: In a Venn diagram, the difference between two sets is represented by shading the region of set A that does not overlap with set B.

The difference between two sets is a useful concept in set theory and allows for the analysis of elements that are unique to a particular set. It helps to determine the unique elements in a given set when compared to another set.

Let's consider an example to illustrate the difference between two sets.

Suppose we have two sets:

A = {1, 2, 3, 4, 5}

B = {3, 4, 5, 6, 7}

To find the difference A - B, we need to determine the elements that are in set A but not in set B.

A - B = {x : x ∈ A and x ∉ B}

Using the given sets, we can determine the difference:

A - B = {1, 2}

Therefore, the difference between set A and set B, denoted as A - B, is {1, 2}. It includes all elements that are present in set A but not in set B.

In this example, the elements 1 and 2 belong to the difference set A - B, as they are in set A but not present in set B.

Symmetric difference between two sets

The symmetric difference between two sets is an operation in set theory that represents the elements that are present in either of the two sets, but not in their intersection. It is denoted by the symbol "⊕" or "Δ".

Given two sets A and B, the symmetric difference of A and B, denoted as A ⊕ B or A Δ B, is defined as the set of elements that are in A or in B, but not in their intersection.

Mathematically, the symmetric difference of A and B is represented as:

A ⊕ B  =  (A - B) ∪ (B - A)  =  (A ∪ B) - (A ∩ B)  =  (A ∩ B') ∪ (A' ∩ B) 

In other words, it includes all elements that are exclusively present in either A or B, but not in both.

Here are some key points about the symmetric difference between two sets:

1. Exclusive Elements: The symmetric difference includes elements that are exclusively present in either A or B. It excludes elements that are common to both sets.

2. Commutative: Unlike the difference operation, the symmetric difference is commutative. This means that A ⊕ B is the same as B ⊕ A. The result is independent of the order of the sets.

3. Empty Intersection: If A and B have no elements in common (A ∩ B = ∅), then the symmetric difference is equal to the union of the sets: A ⊕ B = A ∪ B.

4. Venn Diagram: In a Venn diagram, the symmetric difference between two sets is represented by shading the regions that are exclusively occupied by either A or B.

The symmetric difference between two sets is a useful concept in set theory and allows for the analysis of elements that are unique to either set. It helps to determine the elements that are present in either set but not in their intersection.

Example 1: If A = {1,2,3,4,5,6} , B = {1,4,8,9,10} then A⊕B = _________

A ⊕ B  =  (A - B) ∪ (B - A)

A ⊕ B  =  {2,3,5,6} ∪ {8,9,10}

A ⊕ B  =  {2,3,5,6,8,9,10}

Example 2: Let A and B are two non empty sets and A is a proper subset of B. If n(A) = 5, then minimum possible value for n(A ⊕ B) = _________

Given that A is a proper subset of B, it means that A is a subset of B but not equal to B. Therefore, there exist elements in B that are not in A.

We are also given that n(A) = 5, which means that A has 5 elements.

To find the minimum possible value for n(A ⊕ B), we need to consider the scenario where A and B have the least overlap.

Since A is a proper subset of B, the minimum possible value for n(A ⊕ B) occurs when B contains exactly the same 5 elements as A, plus at least one additional element that is not in A.

In this scenario, the symmetric difference A ⊕ B will include the additional element that is present in B but not in A.

Therefore, the minimum possible value for n(A ⊕ B) is 1.

Hence, the minimum possible value for n(A ⊕ B) is 1 when n(A) = 5 and A is a proper subset of B.