How Many Symmetric Relations Are There in a Set of n Elements?

Understanding the concept of symmetric relations is crucial in the field of discrete mathematics and combinatorics. A symmetric relation on a set is one where if an ordered pair (a, b) is in the relation, then the ordered pair (b, a) must also be in the relation. In this article, we delve into determining the number of such relations possible in a set with n elements.

Introduction to Symmetric Relations

A relation on a set can be represented as a subset of the Cartesian product of the set with itself. For a set S with n elements, the Cartesian product S x S contains n2 ordered pairs. Let's explore how many symmetric relations can be formed from this Cartesian product.

Diagonal and Off-Diagonal Pairs

To determine the number of symmetric relations, we need to consider two types of pairs: diagonal pairs and off-diagonal pairs.

Diagonal Pairs: For each element a in S, the pair (a, a) is a diagonal pair. Since a symmetric relation can include or exclude the pair (a, a) independently, there are 2n ways to choose these diagonal pairs. Off-Diagonal Pairs: For each pair of distinct elements (a, b) where a ≠ b, if (a, b) is included in the relation, then (b, a) must also be included for the relation to be symmetric. This results in 2 choices for each distinct pair: include neither (a, b) nor (b, a), or include both (a, b) and (b, a). The number of distinct pairs (a, b) where a ≠ b is given by n choose 2, which equals (n(n-1))/2.

To find the total number of symmetric relations, we combine the choices for diagonal pairs and off-diagonal pairs:

Number of symmetric relations 2n × 2(n(n-1)/2) 2n (n(n-1)/2) 2(n^2 - n n)/2 2(n^2 - 1)/2.

Explanation with an Example

Consider a set A {1, 2, 3, ..., n}. The Cartesian product A × A consists of n2 ordered pairs. There are n diagonal pairs (i.e., (1,1), (2,2), ..., (n,n)), and (n2 - n) off-diagonal pairs.

The off-diagonal pairs can be paired as follows: [xy, yx] for each distinct pair (x, y). There are (n2 - n)/2 such pairs.

Considering the choices for each diagonal pair and each off-diagonal pair, the total number of symmetric relations can be calculated as:

Number of symmetric relations 2n × 2(n2 - n)/2 2(n^2 - n n)/2 2(n^2 - 1)/2.

Conclusion

In summary, the number of symmetric relations in a set of n elements is given by the formula 2(n^2 - 1)/2. This formula helps in understanding the combinatorial nature of symmetric relations and can be applied in various areas of discrete mathematics and computer science.