Sets and Relations

Sets and relations, including their functions, deal with the most basic yet significant mathematical operations. These are the foundations for concepts like integral and differential calculi. Essentially, a set is a grouping of distinct masses of certain characters. The operations on sets establish the relations between two or more sets present within the universal set. Set operation examples and types include operations performed on the defined group of set items. Operations on sets are similar to the usual mathematical operations, including addition, subtraction, multiplication, and division.

Sets 

• A set is a collection of distinct items or characters. 
• Commas separate items within sets.
• The items of a set are enclosed within curly brackets.
• The alphabet used to indicate a set is always capital.
• An example of a set:

A = {a, b, c, d,….f}

Operations on sets

Operations on sets are similar to the basic operations of mathematics. Operations on sets are of four basic types:

• Union
• Intersection
• Complement 
• Difference 

Sets and relations

• The relations between sets establish how one set is connected to the other. 
• The relation (R) of a non-void Set A to a non-void Set B is a subset of Set A × B. 
• The subset portrays the relation between sets A and B.
• The elements from the first set are termed domains, and those from the second set are termed the range of the relation R.
• For example, if R is the relation between two sets X = {a, b ,c} and Z = {g, h ,i},

Set R = {(a, g), (b, h), (c, i)}, where,

The domain of relation R = {a, b ,c} and

The range of relation R = {g, h, i}

Sets and relations types

The relations or connections between two sets can be of various types. They are as follows.

Empty relation 

• Considering sets A and B, if they have no relation to each other, they have no elements in set relation R. These are empty relation sets.
• Example: Set A is the set of trees in a mangrove forest, and Set B is trees in a desert. They have no relation with each other. Thus, they are a set with no elements in the relation R.

Universal relation 

• The universal set is also called a full relation set.
• Every element in one set is completely related or similar to the other set.
• Example: Set A containing numbers {4, 9, 16, 25} and Set B containing {22, 32, 42, 52} have a complete relation R with each other.

Identity relation 

• In such a relation R, the elements in one set are identical to themselves. 
• Example: If Set A is {a, c, e, g}, its identity relation set is {(a, a),(c, c),(e, e),(g, g)}.
• It is like throwing two dice simultaneously and getting the probability of obtaining the same numerical values on both dice. 

Inverse relation 

• The set relation of the considered set is the inverse of the same set itself.
• Example: If Set A = {(1, 2),(3, 4),(5, 6)}, its inverse is A-1 = {(2, 1),(4, 3),(6, 5)}

Reflexive relation 

• In a reflexive relation, all the elements of the considered set relate to themselves in all possible probabilities.
• Example: If Set A = {a, b}, the reflexive set relation is R = {(a, a),(b, b),(a, b),(b, a)}

Symmetric relation 

• The elements in the relation set are symmetric with the considered set.
• Example: If Set A = {a, b}, the relation set R = {(a, b),(b, a)}

Transitive relation 

• In a transitive relation, the elements are transitive to each other.
• Example: If (a, b) belongs to R and (b, c) belongs to R, then (a, c) also belongs to R.

Solved examples

Let’s make the above discussions clear and also check your understanding. Here are some solved examples for you.

• Consider Set A = {a, e, i, o, u}. What is the identity set of Set A?
• In an identity set, the pairs of elements are identical to the elements in the given set.
• Thus, the identity relation of Set A = {(a, a), (e, e), (i, i), (o, o), (u, u)}
• Consider the set relation R = {(a, z), (b, y), (c, x), (d, w)}. What is the domain and range of the set relation?
• The domain of set relation R = {a, b, c, d}
• The range of set relation R = {z, y, x, w}
• Consider the following sets.

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

P = {1, 3, 5, 7}

Q = {2, 4, 6, 8}

What is the complement of Set Q?

Set Q = {2, 4, 6, 8}

Set Q’ = {1, 3, 5, 7, 9, 10}

Conclusion 

A set is a group of distinct characteristics enclosed inside curly brackets and separated by commas. Operations on sets are similar to basic mathematical functions. Set operation examples include union, intersection, complement, and difference of sets. Sets and relations show how the sets are connected or related to each other or themselves. The set relation is denoted by R. Sets and relations are of various types, classified based on the nature of the relation of one set to the other.