1. What a Symmetric Relation Means
A symmetric relation is one where every connection works both ways. If an element a is related to an element b, then b must also be related to a. So whenever (a,b) appears in the relation, the pair (b,a) must also appear.
2. Formal Condition
The condition for a relation R on a set A to be symmetric is:
2.1. Definition
(a,b) \in R \; \Rightarrow \; (b,a) \in R
3. Examples of Symmetric Relations
Here are relations that satisfy the symmetric condition:
3.1. Example 1
Let A = \{1,2,3\}. Consider:
R = \{(1,2),(2,1),(2,3),(3,2)\}
Every pair has its reverse, so R is symmetric.
3.2. Example 2
A relation can be symmetric even with self-pairs:
R = \{(1,1),(2,2),(3,3)\}
Self-pairs do not break symmetry.
4. How to Check Symmetry
To check if a relation is symmetric, scan each ordered pair in the relation. For every (a,b), check whether (b,a) is also present. If even one reverse pair is missing, the relation is not symmetric.
4.1. Illustration
For A = \{x,y,z\}:
R = \{(x,y),(y,x),(x,z),(z,x)\}
Since all reverse pairs are included, this relation is symmetric.
5. Non-Examples (Not Symmetric)
When a relation contains a pair (a,b) but not the pair (b,a), it fails to be symmetric.
5.1. Example
R = \{(1,2),(2,3)\}
The reverse pairs (2,1) and (3,2) are missing, so R is not symmetric.
6. Situations That Naturally Form Symmetric Relations
Symmetric relations appear when a relationship naturally works in both directions. Some examples include:
- “is a sibling of”
- “has the same colour as”
- “is connected to” in undirected networks