Symmetric Relations

A relation is a subset of the Cartesian product of two sets, connecting elements through ordered pairs. In each ordered pair, the first element comes from the domain, and the second from the range.

In other words, it connects elements of one set to elements of another through ordered pairs (a, b), where 'a' is from the domain and 'b' is from the range.

• A = {x, y}

• B = {3, 4, 5}

• A relation between A and B could be R = {(x, 5), (y, 3)}
   

A symmetric relation on a set X means that whenever an ordered pair (a, b) is in R, then the pair (b, a) must also be included in R. Every connection between two elements will be mutual; if a relates to b, then b relates to a.

Examples

1. "Is equal to" is an example of symmetric relations, such as $1 = 100 cents, and 100 cents = $1.
    
2. "Is friend of " is also a symmetric relation.

In mathematics, relations define connections between elements of sets using ordered pairs.

• A relation is said to be symmetric if, whenever one element is related to another, the reverse is also true.
   
• A relation is asymmetric if, whenever a is related to b, b is never related to a.
  Additionally, elements cannot relate to themselves.

• A relation is antisymmetric if (a, b) ∈ R and (b, a) ∈ R imply a = b.

Here are some important points to remember. 

1. Asymmetric ⇒ Antisymmetric (and implies irreflexive).
 If xRy, then yRx cannot hold, so antisymmetry is satisfied.

2. Antisymmetric relations allow (a, a) ∈ R (e.g., ≤), but asymmetric relations do not.

3. Symmetric and asymmetric are mutually exclusive, except for the empty relation.

4. Symmetric and antisymmetric relations coincide only in trivial cases, such as the identity relation {(x, x) | x ∈ X}.

A relation R on a set A is symmetric if:
\(∀a, b∈A,  (a, b)∈R  ⟹  (b, a)∈R.\)

Equivalently, R = R^-1(its inverse)

1. Inverse equals itself:
 R^-1 = R.

2. Closed under set operations:
If R₁ and R₂ ​are symmetric, then so are R₁ ∪ R₂ and R₁ ∩ R₂​.

3. Matrix representation is symmetric:
Its adjacency matrix satisfies M = M^T

4. Digraphs have paired edges:
Every directed edge a→b is accompanied by b→a.

5. Extremes are symmetric:
The empty relation and the universal relation (all pairs) are both symmetric.
 

A relation is both symmetric and antisymmetric only in trivial cases, such as the identity relation {(x, x) | x ∈ X}.

Number of Symmetric Relations Formula

Number of symmetric relations = \(2^ { \ n \ ({ n \ + \ 1 \over 2})}\)

Where

• n is the number of elements in the set,
• Let N denote the number of symmetric relations.

Let's practice this using an example.
 

Example:  Set A = {1,2,3}:

Explanation: Here n = 3,

so, number of symmetric relations = \(2^ { \ 3\ ({ 3 \ + \ 1 \over 2})} = 2 ^ { 3 \times 2} = 2 ^ 6 = 64\)
 

Practice Problem: Similarly, find the number of symmetric relation of set B = {3, 2, 5, 3} by yourself for practice.

To check if a relation R on a set A is symmetric, follow these steps:

1. List all ordered pairs in R, such as (a, b).
    
2. For each pair, verify that the reverse (b, a) is also in R.
    
3. If every pair has its reverse, the relation is symmetric.
    
4. If any pair (a, b) ∈ R lacks its reverse (b, a) ∈ R, the relation is not symmetric.

Let's practice this using some problems.

Problem 1:
Let R = { (1, 2), (2, 1), (3, 4), (4, 3) }.
Each pair has its reverse, so R is symmetric.

Problem 2:
Let R = { (1, 3), (3, 4), (4, 3) }.
The pair (1, 3) does not have its reverse, so R is not symmetric.

To make symmetric relations easy and simple, here are a few quick tips and tricks:

1. To check the symmetry, just flip the pair, and check if it exists in the relations.
    
2. To find the transpose of a relation matrix, write the rows and columns and columns as rows.
    
3. Use real life analogy for better understanding.
    
4. Memorize the formula for calculating the number of symmetric relations, which is \(2^ { \ n \ ({ n \ + \ 1 \over 2})}\)
    
5. Remember, every equivalence relation is symmetric, however the vice versa is not true.
    

Parent Tip: Relate symmetric relations to real life relations to help your child understand better. Encourage your child to practice. 

Symmetrical relations are prevalent in various aspects of our daily lives, from social interactions to mathematical concepts. Let’s see some of their real-life applications.

1. Nature: Both butterfly wings have the same shades, designs and color; if one side has any pattern, then the other side will have the same. It is an example of symmetric relations.
   
    
2. Biology: The relation ‘has similar DNA’ between two organisms is symmetric.
   For example, if a person has two children, the DNA's of both child are similar.
   
    
3. Social Networks: In Neuroscience, the brain structure shows that the symmetry in both hemispheres deals with similar types of work; if one hemisphere is damaged, the other will also be affected.  
   
    
4. Technology: Game design and graphics of computer simulation are used in mesh modelling, and character design for reduced and balanced compilation. 
   
    
5. Art & Design: The software used in parametric design ensures symmetrical balance in 3D forms while building furniture, chairs, bench, or any architectures. 
   For example, if you are drawing a chair, you need to draw all legs of chairs of the same height.