Algebraic Properties of Sets

Introduction

Algebraic properties are some rules that are needed to solve problems. Algebraic properties of sets contain various rules that one needs to remember. Based on these rules, various mathematical problems and their solutions are there. Using these rules and properties, you will be able to understand the logic behind many set problems.

Concepts of sets in mathematics

The notion of sets in mathematics offers many properties and operations on collections of elements. Sets are the basis of many forms of data analytics. It is useful in the classification and organization of data. 

Fundamentals

The algebra of sets is the set-theoretic parallel of the algebra of numbers. As we know, algebraic addition and multiplication properties are associative and commutative, in the same set union and intersection are. In algebra, we have a relation “less than or equal”. It is reflexive, antisymmetric, and transitive. In the same way, we have a set relation of ”subset”. 

It is the same as the algebra of the set-theoretic operations of intersection, union, and complementation, and the relations of equality and inclusion.

Sets in real life

Let us now see how sets are used in day-to-day life. Do you have a favourite dress? You must have. Let us assume it is a skirt from a branded shop. The skirts may have a variety depending upon their design. So, this is called a set in real life. Set is nothing but a collection of items of the same category. Technically, a set is the collection of particular elements. Let’s understand it correctly.

Set Theory 

There are some significant features of set theory which are mentioned below- 

• We make a set by placing items of a single entity together. This will form a set. 
• The Principle of Extension in sets tells that the set is known by its objects. Thus, sets A and B are equal only if all the elements should intersect, and none of them has any unique elements which are not present in the other set.
• There may or may not be any specific relationship between an object and its set. It is called a membership relationship. 

The fundamental properties of a set

Now, let’s focus on the fundamental properties of sets. Suppose P and Q be two sets, then we can define a set P intersection Q denoted by P⋂Q, whose elements consist of all the common elements of P and Q. Another set, P union Q, denoted by P⋃Q, is the set that contains all the elements of P and Q. A complement denoted by P’ is the set of numbers of the universal set U, except for the elements of P, the set which does not contain any members is known as the null set. It is usually denoted by ∅. The operations and properties of sets satisfy many identities.

Property 1. Commutative property

Union and intersection operation satisfy the commutative property, which means that p+q= q+p in algebraic terms. In terms of set, it is :

A⋃B = B⋃A

A⋂B = B⋂A

Property 2. Associative property

Again, both union and intersection fulfil the associative property of sets. In simple terms, it means that (p+q)+r is the same as p+(q+r). In terms of sets, it is as follows:

(A⋃B)⋃C = A⋃(B⋃C)

(A⋂B)⋂C = A⋂(B⋂C)

Property 3. Distributive property

Same as before, union and intersection both satisfy the distributive property of set. It is as follows:

A⋂(B⋃C) = (A⋂B)⋃(A⋂C)

A⋃(B⋂C) = (A⋃B)⋂(A⋃C)

Property 4. Identity

Identity property means that a+0 is always equal to a. 

A⋂U = A

A⋃∅ = A

Property 5. Idempotent

A⋃A = A

A⋂A = A

Property 6. Complement

A⋂A’ = ∅

A⋃A’ = U

Property 7. De Morgan’s Laws

 For any two finite sets A and B;

(i) A – (B ∩ C) = (A – B) U (A – C)

(ii)A – (B U C) = (A – B) ∩ (A – C)

De Morgan’s Laws can also be written as:

(i)(A ∩ B)’ = A’ U B’

(ii) (A U B)’ = A’ ∩ B’

The principle of duality

All the properties that are stated above are one of a pair of identities such that each can be changed into the other by interchanging ∪ and ∩, and also Ø and U. This is the principle of duality.

More laws of the algebra of sets:

Let’s see some more properties of sets which can be used sometimes. 

Property 8. 

For any two finite sets P and Q;

(i)(P – Q) ∩ Q = ∅

(ii) Q – P = Q ∩ P’

(iii) P ⊆ Q ⇔ Q’ ⊆ P’

(iv) (P – Q) U Q = P U Q

(v) P – Q = P ∩ Q’

(vi)P – Q = P ⇔ P ∩ Q = ∅

(vii) (P – Q) U (Q – P) = (P U Q) – (P ∩ Q)

Property 9.

 For any three finite sets P, Q and R;

(i) P – (Q ∩ R) = (P – Q) U (P – R)

(ii) P – (Q U R) = (P – Q) ∩ (P – R)

(iii) P ∩ (Q – R) = (P ∩ Q) – (P∩ R)

(iv) P ∩ (Q U R) = (P ∩ Q) U (P ∩ R)

Conclusion

Algebraic properties of sets are a set of some rules and properties which are used to define a set of elements. This is all you need to study about a set and its properties. The main properties of a set are distributive, commutative, and associative properties. All these are used to prove identities in sets.