Operations on Sets

• Union of Sets
• Intersection of Sets
• Addition theorem (Inclusion-Exclusion principle)

Union of Sets:

In set theory, the union of two or more sets is a new set that contains all the distinct elements from the given sets. The union operation is denoted by the symbol "∪".

Here are the key points regarding the union of sets:

1. Definition: The union of sets A and B, denoted as A ∪ B  or  A + B  or  at least one of A or B  or  A OR B , is the set that contains all elements that are in A, in B, or in both A and B. Formally, x ∈ (A ∪ B) if x ∈ A or x ∈ B.
2. Elements in the Union: The union of sets contains all the distinct elements from the individual sets. If an element is present in multiple sets, it appears only once in the union.
3. Example: Let's consider two sets: A = {1, 2, 3} and B = {3, 4, 5}. The union of A and B, denoted as A ∪ B, would be {1, 2, 3, 4, 5}. Note that the element "3" appears only once in the union, even though it is present in both sets.
4. Union of Multiple Sets: The union operation can be extended to more than two sets. For example, if we have sets A, B, and C, the union of these sets, denoted as A ∪ B ∪ C, is the set that contains all the distinct elements from A, B, and C.
5. Commutative Property: The union operation is commutative, which means that the order of the sets does not affect the result. A ∪ B is equivalent to B ∪ A.
6. Associative Property: The union operation is associative, which means that when performing the union of three or more sets, the grouping of the sets does not affect the result. For example, (A ∪ B) ∪ C is equivalent to A ∪ (B ∪ C).

Some more property:

• Ø ⋃ A = A
• A ⋃ A = A
• A ⋃ U = U                   (Where , U = Universal Set)
• If we have 'n' Set

A₁, A₂, A₃, A₄, A₅, A₆, A₇, A8,.............,A_n-1, A_n

then,

 A₁ ⋃ A₂ ⋃  A₃ ⋃  A₄ ⋃ A₅ ⋃ A₆ ⋃  A₇ ⋃  A8 ⋃ ............. ⋃ A_n-1 ⋃  A_n =  

The union of sets is a fundamental operation in set theory and is used to combine sets, find the combined elements, and express the concept of inclusiveness or combination. It is an essential concept in various branches of mathematics and has applications in logic, algebra, probability theory, and more.

Intersection of Sets:

In set theory, the intersection of two or more sets is a new set that contains only the elements that are common to all of the given sets. The intersection operation is denoted by the symbol "∩".

Here are the key points regarding the intersection of sets:

1. Definition: The intersection of sets A and B, denoted as A ∩ B  or  A • B  or A and B  or  Both A and B , is the set that contains all elements that are present in both A and B. Formally, x ∈ (A ∩ B) if x ∈ A and x ∈ B.
2. Elements in the Intersection: The intersection of sets contains only the elements that are common to all the sets being intersected. If an element is not present in any one of the sets, it is not included in the intersection.
3. Example: Let's consider two sets: A = {1, 2, 3} and B = {3, 4, 5}. The intersection of A and B, denoted as A ∩ B, would be {3}, since it is the only element that appears in both sets.
4. Intersection of Multiple Sets: The intersection operation can be extended to more than two sets. For example, if we have sets A, B, and C, the intersection of these sets, denoted as A ∩ B ∩ C, is the set that contains all the elements that are present in all three sets.
5. Commutative Property: The intersection operation is commutative, which means that the order of the sets does not affect the result. A ∩ B is equivalent to B ∩ A.
6. Associative Property: The intersection operation is associative, which means that when performing the intersection of three or more sets, the grouping of the sets does not affect the result. For example, (A ∩ B) ∩ C is equivalent to A ∩ (B ∩ C).

Some more property:

• Ø ∩ A = Ø
• A ∩ A = A
• A ∩ U = A                   (Where , U = Universal Set)
• If we have 'n' SetA₁, A₂, A₃, A₄, A₅, A₆, A₇, A8,.............,A_n-1, A_n

then

 ,

A₁ ∩ A₂ ∩  A₃ ∩  A₄ ∩ A₅ ∩ A₆ ∩  A₇ ∩  A8 ∩ ............. ∩ A_n-1 ∩  A_n =

The intersection of sets is a fundamental operation in set theory and is used to identify the common elements among different sets. It helps in finding the shared properties or characteristics of objects within a given context. The concept of intersection is widely applied in mathematics, statistics, logic, and other fields.

Addition theorem | Inclusion-Exclusion principle

In set theory, the addition theorem (also known as the inclusion-exclusion principle) is a formula that allows us to calculate the size of the union of multiple sets by considering the sizes of the individual sets and their intersections. The addition theorem is expressed as follows:

For two sets:
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

For three Sets:
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ B) - n(A ∩ B) + n(A ∩ B ∩ C)

For n number of Sets:

or

 
n(A ∪ B ∪ ... ∪ N) = n(A) + n(B) + ... + n(N) - n(A ∩ B) - n(A ∩ C) - ... - n(M ∩ N) + n(A ∩ B ∩ C) + ... + (-1)^(n-1) n(A ∩ B ∩ ... ∩ N),

or

|A ∪ B ∪ ... ∪ N| = |A| + |B| + ... + |N| - |A ∩ B| - |A ∩ C| - ... - |M ∩ N| + |A ∩ B ∩ C| + ... + (-1)^(n-1) |A ∩ B ∩ ... ∩ N|,

where |X| and n(X) represents the cardinality (number of elements) of set X and (-1)^(n-1) denotes alternating signs.

In simpler terms, the addition theorem states that to find the size of the union of multiple sets, we sum the sizes of each set, then subtract the sizes of the pairwise intersections, add back the sizes of the triple intersections, and so on.

This principle accounts for overcounting or undercounting elements when taking the union of multiple sets. By including or excluding the sizes of intersections appropriately, we obtain an accurate count of elements in the union.

The addition theorem is particularly useful when dealing with situations where sets overlap or intersect. It provides a systematic way to calculate the size of the union, considering both shared and unique elements across multiple sets.

Here are a few examples that demonstrate the application of the inclusion-exclusion principle:

Example 1: 

Consider three sets A, B, and C, with the following sizes: |A| = 20, |B| = 25, |C| = 30. The sizes of the pairwise intersections are: |A ∩ B| = 10, |A ∩ C| = 15, |B ∩ C| = 8. Finally, the size of the triple intersection is: |A ∩ B ∩ C| = 5. Using the inclusion-exclusion principle, we can calculate the size of the union of the three sets as:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

              = 20 + 25 + 30 - 10 - 15 - 8 + 5

              = 47

Therefore, the size of the union of sets A, B, and C is 47.

Example 2: 

Let's consider four sets P, Q, R, and S, with the following sizes: |P| = 50, |Q| = 60, |R| = 70, |S| = 80. The sizes of the pairwise intersections are: |P ∩ Q| = 25, |P ∩ R| = 35, |P ∩ S| = 40, |Q ∩ R| = 15, |Q ∩ S| = 30, |R ∩ S| = 20. The size of the triple intersections are: |P ∩ Q ∩ R| = 10, |P ∩ Q ∩ S| = 15, |P ∩ R ∩ S| = 12, |Q ∩ R ∩ S| = 8. Finally, the size of the quadruple intersection is: |P ∩ Q ∩ R ∩ S| = 5. 

Using the inclusion-exclusion principle, we can calculate the size of the union of the four sets as:

|P ∪ Q ∪ R ∪ S| = |P| + |Q| + |R| + |S| - |P ∩ Q| - |P ∩ R| - |P ∩ S| - |Q ∩ R| - |Q ∩ S| - |R ∩ S|

                  + |P ∩ Q ∩ R| + |P ∩ Q ∩ S| + |P ∩ R ∩ S| + |Q ∩ R ∩ S| - |P ∩ Q ∩ R ∩ S|

                  = 50 + 60 + 70 + 80 - 25 - 35 - 40 - 15 - 30 - 20 + 10 + 15 + 12 + 8 - 5

                  = 85

Therefore, the size of the union of sets P, Q, R, and S is 85.

The inclusion-exclusion principle allows us to calculate the size of the union of sets accurately by accounting for shared elements and avoiding double-counting. It can be extended to any number of sets and is a valuable tool in counting and combinatorial problems involving sets.

Example 3: 

In class 11, there are 30 students in mathematics class and 20 students in Biology class. find the number of students which are in mathematics class or in Biology class in following cases:
A) Two classes are conducted at the same time. 
B) Two classes are conducted at different times and 5 students study both subjects.

A) To find the number of students who are either in the mathematics class or in the biology class when the two classes are conducted at the same time, we need to calculate the size of the union of the two classes.

Given:

Number of students in the mathematics class = 30

Number of students in the biology class = 20

To find the number of students in either class, we calculate the union of the two classes:

Number of students in mathematics class or biology class = |Mathematics class ∪ Biology class|

Using the formula for the union:

|Mathematics class ∪ Biology class| = |Mathematics class| + |Biology class| - |Mathematics class ∩ Biology class|

Since the classes are conducted at the same time, there are no students who can be in both classes simultaneously. Therefore, the intersection of the two classes, |Mathematics class ∩ Biology class|, is 0.

Substituting the values:

Number of students in mathematics class or biology class = 30 + 20 - 0 = 50

Therefore, when the two classes are conducted at the same time, there are 50 students in total who are either in the mathematics class or in the biology class.

B) To find the number of students who are either in the mathematics class or in the biology class when the two classes are conducted at different times, considering that 5 students study both subjects, we can use the same formula for the union of two sets.

Given:

Number of students in the mathematics class = 30

Number of students in the biology class = 20

Number of students studying both subjects = 5

To find the number of students in either class, we calculate the union of the two classes:

Number of students in mathematics class or biology class = |Mathematics class ∪ Biology class|

Using the formula for the union:

|Mathematics class ∪ Biology class| = |Mathematics class| + |Biology class| - |Mathematics class ∩ Biology class|

Substituting the values:

|Mathematics class ∪ Biology class| = 30 + 20 - 5 = 45

Therefore, when the two classes are conducted at different times, with 5 students studying both subjects, there are a total of 45 students who are either in the mathematics class or in the biology class.

Example 4: 

Let 'Z' be the set of all integers and A= {(a, b): a^2 + 3b^2 =28, a and b from Z} , B = {(a, b): a>b, a and b from Z} ,  then n(A ∩ B) = ___________

We have A = {(a, b): a^2 + 3b^2 = 28, a and b from Z}. This is the set of all integer solutions to the equation a^2 + 3b^2 = 28.

Similarly, we have B = {(a, b): a > b, a and b from Z}. This is the set of all integer pairs (a, b) such that a is greater than b.

To find n(A ∩ B), we need to find the number of elements that are common to both A and B.

Let (x, y) be an element in A ∩ B. Then we have:

(x, y) ∈ A and (x, y) ∈ B

=> x^2 + 3y^2 = 28 and x > y

Solving for x and y in terms of a third variable, we get:

x = ±√(28 - 3y^2) and y = k for some integer k.

Since x > y, we have x = √(28 - 3y^2) for some y such that 0 ≤ y ≤ 3.

Thus, the possible values of (x, y) are:

(5, 1), (-5, 1), (4, 2), (-4, 2), (3, 3), (-3, 3)

Therefore, n(A ∩ B) = 6.

Hence, there are 6 integer pairs that belong to both sets A and B.