Question

How many transitive relations are there on a set with n elements if

a) n = 1
b) n = 2
c) n = 3

Answer 1

The number of transitive relations on a set with n elements is 2^(n(n-1)/2).

For n = 1, there is 1 possible transitive relation.

For n = 2, there are 2 possible transitive relations.

For n = 3, there are 8 possible transitive relations.

Step-by-step explanation:

A transitive relation on a set with n elements is a relation where if a is related to b and b is related to c, then a is related to c.

To determine the number of transitive relations, we can use the fact that each pair of distinct elements in the set can either be related or not related, resulting in 2 possible options.

Therefore, the number of transitive relations on a set with n elements is 2^(n(n-1)/2).

a) For n = 1, there is only 1 element in the set, so there is only 1 possible transitive relation.

b) For n = 2, there are 2 elements in the set, resulting in 2^(2(2-1)/2)

= 2^1

= 2 possible transitive relations.

c) For n = 3, there are 3 elements in the set, resulting in 2^(3(3-1)/2)

= 2^3

= 8 possible transitive relations.