Symmetric Property

In algebra, the symmetric property expresses that if one element in a set is related to another, then the second element is also related to the first in the same manner. This concept can be seen in several forms in mathematics, including:
Symmetric property of equality states that if x = y, then y = x

Symmetric property of congruence states: If in a set, one geometric figure is congruent to another, then the other figure is also congruent to the first one.

Symmetric property of relations says:  If a relation R is symmetric, then aRb ⇔ bRa, for all a, b.

Symmetric properties of matrices suggest:  If a matrix A is symmetric, then it is equal to its transpose A = A^T
 

In mathematics, a relation shows how two or more values in a set are associated with each other. In an ordered pair with related elements, the first element is the domain and the second is the range. 

For example, consider the following sets:
P = {a, b}
Q = {1, 2, 3}

A relation R from P to Q could be:
R = (a, 2) (b, 1)
This indicates that a is related to 2, and b is related to 1.
This relation is a subset of the Cartesian product of two sets, P × Q.

For a relation to be symmetric, each ordered pair in the given relation must satisfy the given condition:
(a, b) R (b, a) R

For example: 

Question: Consider a set A = 1, 2, 3 and a relation R on A defined as:
R = (1, 2), (2, 1), (2, 3), (3,2)
Now, let’s check if R is symmetric.
Solution: 
(1, 2) is in R → Is (2, 1) in R?  Yes

(2, 1) is in R → Is (1, 2) in R?   Yes

(2, 3) is in R → Is (3, 2) in R?   Yes

(3, 2) is in R → Is (2, 3) in R?  Yes

Since all pairs have a reverse pair in the relation, R is symmetric.
 

Relations in mathematics are used as a way to show the connection between the elements of two sets. Among these relations are asymmetric and symmetric relations that differ from each other in the following ways:

The formula for a total number of symmetric relations with n elements is:

N = 2^[n(n+1)]/2

Where, 
N is the number of symmetric relations, and n is the number of elements in the set.
Explanation:
The total number of possible ordered pairs from set A is: n² =n × n

In a symmetric relation, if the pair (a, b) is included, then (b, a) must also be included. So, instead of selecting pairs freely, we select:

• n diagonal elements (i.e., pairs like (a, a)): each can be included or not → 2 options per element.
   
• n(n – 1)/2 off-diagonal element pairs, like (a, b) and (b, a): these come as pairs, and each pair either appears together or not at all → again 2 options per such pair.

Therefore, the total number of symmetric relations is:
2ⁿ × 2^[n(n-1)]/2 = 2^[n(n+1)]/2

For example: For a set A = 1, 2, n = 2

N = 2^[n(n+1)]/2 = 2³ = 8

So, for a 2-element set, there are 8 symmetric relations.
 

As one learns about the symmetric property, it is also essential to understand its applications in real-world scenarios. Some of these are listed below.

• Legal and business agreements: The symmetric property establishes that if a contract states that A owes B ‘x’ amount of money, then B is owed ‘x’ amount from A. This symmetric understanding ensures clarity in responsibilities and rights in legal or business transactions.
• Currency Conversion: The symmetric property helps in converting currencies both ways. For instance, if $1 = ₹83, then ₹83 = $1.
• Mirror Design in Architecture and Fashion:  Designers use symmetric properties to create balanced patterns in architecture and fashion. This means if one side (left) mirrors the other (right), making the design look the same from both directions.
• Explaining Prescriptions to Patients: Doctors prescribe medicines in dosages. Using the symmetric property, they can explain the amount to their patients. For example, 5 mL = a teaspoon, so one teaspoon of medicine will be equal to 5 mL.
• Assigning variables in coding : In coding, the symmetric property is used for comparing variables in search algorithms, data matching or conditional logic.