Relation-Inverse Relation

If R is a relation from a set A into another set B, then by interchanging the first and second coordinates of ordered pairs of the relation( R), then a new relation is formed from B to A. This relation is called Inverse Relation.

Set of ordered pairs satisfying the given condition is called Relation( R). If we are given two sets, A and B, such that

A= {1, 2, 3} and B= {a, b}

Cartesian product of given sets A and B will be the set of ordered pairs of set A and set B, where the first element of the ordered pair belongs to set A and the second element of the ordered pair belongs to set B.

A*B = {(1,a), (1,b), (2,a), (2,b), (3,a),(3,b)}

We know that any subset of the Cartesian product of given sets (A and B here) is a relation( R).

Therefore, we can find out inverse of a relation as:

R = {(b, a) : b ∈ B and a ∈ A }

Therefore, Inverse of Relation here will be:

{(a,1), (a,2), (a,3), (b,1), (b,2), (b,3)}

Inverse Relation Mathematical notation and Formula:

The inverse of a given relation R is denoted by R^-1.

Formula for inverse relation(R^-1) = {(b, a): b ∈ B and a ∈ A}. i.e.,

We can also show the inverse relation as:

If (a, b) ∈ R, then (b, a) ∈ R^-1

Meaning the sequence of first and second elements in ordered pairs will interchange for all the ordered pairs.

R ⊆ A*B, then R^-1 ⊆ B*A.

Note that:

• The number of ordered pairs in the original relation and the inverse relation are the same.
• The first element of the ordered pairs of original relation has now become the second element of the inverse relation.
• The second element of the ordered pairs of the original relation has changed to the first element while calculating the inverse of a relation.

Let us explore the inverse relation with some illustrations.

Example:

Given two sets P = {Tea, Coffee} and Q = {Alex, Julie, Inaya, Dee}.

And R = {(p,q) : All are tea drinkers}

Solution:

R = {(Tea, Alex), (Tea, Julie), (Tea, Inaya), (Tea, Dee}

This implies that

R^-1 = {(Alex, Tea), (Julie, Tea), (Inaya, Tea), (Dee, Tea)}

From our solution, we can observe that the number of ordered pairs in the original relation or given relation is 4 and the same number is obtained by taking an inverse of the given relation.

The first elements of ordered pairs in the original relation were {Tea}, which eventually became the second element after we applied the Inverse operation on the given relation( R).

Same id true for all the second elements of given original relation( R). Second elements in the original relation were {Alex, Julie, Inaya, Dee}. However, as we calculated the inverse of the given relation( R), all the second elements eventually got converted into first elements of the ordered pairs of the inverse relation(R^-1).  

For the given scenario, we can write our solution mathematically as below:

If (p, q) ∈ R ⇔ (q, p) ∈ R^-1

Therefore,

For given relation R = {(p, q) : p ∈ P, q ∈ Q}

Then inverse relation R^-1 = {(q, p) : q ∈ Q, p ∈ P}

Range and Domain of Inverse Relation

We know that Domain is the set of all the first elements of the ordered pairs in relation to R, whereas Range is the set of all the second elements of the ordered pairs in relation to R.

Continuing the same example, If we find out the Range and Domain for the given relation( R), then we can also find out the Range and Domain for the Inverse Relation(R^-1  ) and can also establish a connection between the two:

For given relation R = {(Tea, Alex), (Tea, Julie), (Tea, Inaya), (Tea, Dee}

Domain of R = {Tea}

Range of R = {Alex, Julie, Inaya, Tea}

Now let’s make the same calculations for inverse relation(R^-1 )

R^-1 = {(Alex, Tea), (Julie, Tea), (Inaya, Tea), (Dee, Tea)}

Domain of R^-1 = {Alex, Julie, Inaya, Tea}

Range of R^-1 = {Tea}

Points to Note:

• Domain of R eventually becomes the Range for Inverse Relation(R^-1) and vice-versa.
• For any empty relation(R ), the inverse of the relation(R^-1  ) will be R.
• For a symmetric relation, the inverse of the relation(R^-1  ) will be R itself.

Conclusion

Inverse relation is nothing but the interchange of elements of ordered pairs of given relation. We can infer the same from its name as well. However, as we applied the operation of Inverse relation, we noticed how associated things are changing for some properties and are constant for others. We noticed the Range and Domain got interchanged with the application of operation while the number of ordered pairs remained unchanged. Some related properties are also noted in association with Inverse Relation.

In this article, we discussed in detail the meaning of inverse relation, its mathematical notation and formulas. We understood all the concepts with the help of a practical example and covered the important notes and observations on inverse relation operations on sets, pairs of sets and functions.