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 like ordered pairs (a, b), where a is in the domain and b is in 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, the reversed pair (b, a) should also be in R. Every connection between two elements will be mutual; if a relates to b, then b relates to a.

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, and elements cannot be related to themselves.
A relation is antisymmetric if (a, b) ∈ R and (b, a) ∈ R imply a = b.

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 = MT

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 = 2n(n-1)2

Where
n = number of elements in the set,

Let N denote the number of symmetric relations.

For a set A = {1,2,3}:

23(3+1)2 = 26 = 64
 

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

List all ordered pairs in R, such as (a, b).

For each pair, verify that the reverse (b, a) is also in R.

If every pair has its reverse, the relation is symmetric. If any pair (a, b) ∈ R lacks its reverse (b, a) ∈ R, the relation is not symmetric.

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

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.

• Nature: Butterfly wings have the same shades and colour; if one side has any pattern, then the other side will have the same.

• Biology – Genetic Similarity: The relation ‘has similar DNA’ between two organisms is symmetric. 

• Social Networks – Neuroscience, the brain structure shows that the symmetry in both hemispheres deals with similar types of work; if one hemisphere is damaged, the other other will also be affected.  

• Technology – Game design and graphics of computer simulation are used in mesh modelling, and character design for reduced and balanced compilation. 

• Art & Design – The software used in parametric design ensures symmetrical balance in 3D forms while building furniture.