intersection of sets and their algebraic properties

Intersection of Sets and Their Algebraic Properties

Introduction

A set is a consolidation of the elements of a group based on a common category. For example, it can be a set of students in class 7, a set of heights of 13-year-old girls, and even a set of cricketers in a particular team. These are applied to any group of numbers or variables, be it simple numbers right to complex scientific results. The sets operate according to 4 different basics – union, intersection, complement, and difference. Every basic set has its algebraic properties for calculation purposes. Similar to the other operations, the intersection of sets and their algebraic properties aid in perfecting the calculations of a set equation.

Set operations

The set operations are the components that establish the relationship between two or more sets and quantitatively categorise their elements. The most basic set operations are, for example, for two sets, P and Q:

• Union -shows the elements that occupy all the considered sets.

(P ∪ Q)

• Intersection shows the elements that are only common to the considered sets.

(P ∩ Q)

• Complement shows the elements that are present in the Universal but not the considered set.

(P’, Q’) 

• The difference shows the unique elements present in a set. It is more like all the elements present in a set except for the elements at the intersection. 

(P – Q) 

Algebraic Properties of Set Operations

All the operations of the set are governed by their algebraic properties. It is very similar to the basic operations on numbers. The algebraic properties of sets include the following.

• Commutative property
• Associative property
• Distributive property
• Identity
• Complement
• Idempotent

Intersection of sets

• The common elements among two more sets are called the intersection of sets. 
• The elements present in the common region are the intersection elements of the sets. 

For example, consider two sets M and P.

• M ∩ P denotes the intersection operation of the sets M and P.
• n(M ∩ P) provides the total number of elements present in the sets M and P intersecting region.

If M = {1,3,5,7} and P = {3,5,7,9}, then the intersection of M and P, 

M ∩ P = {3,5,7} 

And the number of elements present in the intersection of the sets M and P,

n(M ∩ P) = 3

Algebraic properties of set Intersection

The set intersection is operated by the following algebraic properties similar to the other set operations. Consider the sets M, P, and Q.

Commutative property of Set Intersection:

• For the sets M and P, the commutative property is expressed as,

M ∩ P = P ∩ M

• The commutative property of set intersection states that even when the order of sets interchanges, the intersection elements will be the same.

Associative property of Set Intersection:

• For the sets, M, P, and Q, the associative property is expressed as,

(M ∩ P) ∩ Q = M ∩ (P ∩ Q)

• The associative property of set intersection states that, even when the sets grouped under parentheses are interchanged, the intersection elements will be the same.

Distributive property of Set Intersection:

• For the sets, M, P, and Q, the distributive property is expressed as,

M ∩ (P ∪ Q) = (M ∩ P) ∪ (M ∩ Q)

• The distributive property of set intersection states that the value will be the same even when the common set is distributed among the two sets separately. 

Identity property of Set Intersection:

• For the sets M and P, the identity property is expressed as,

U ∩ M = M and U ∩ P = P

• According to the identity property of set intersection, when the set (M) is at an intersection with the universal set (U), the resulting elements will only be from the set (M).
• The set U gives the same set (example – M) as the answer of intersection with the corresponding set. Thus, ‘U’ is the identity of the set intersection.
• 𝜙 is the identity of the set union.

Complement property of Set Intersection:

• For a set M, the complement property is expressed as,

M ∩ M’ = 𝜙 

• As per the complement property of set intersection, the intersection of a particular set with its complement gives null, i.e., no elements.

Idempotent property of Set Intersection:

• For a set M, the idempotent property is expressed as,

M ∩ M = M

• According to the idempotent property of set intersection, an intersection of the set gives the same set.
• It is the same for the union operation too. Union of a set with itself gives the same set.

Conclusion

The intersection of sets and their algebraic properties govern the correctness of an equation while calculating set intersections. Every set operation has properties similar to the algebraic properties of the intersection of sets. These algebraic rules remain similar for the set operations but not the same, always. For example, the identity set of the operations union and intersection differs.