Lecture 6 - Equivalence Relations

Question 1

What is an equivalence relation?

Answer 1

A relation on a set that is reflexive, symmetric and transitive.

Question 2

What does an equivalence relation look like in a digraph?

Answer 2

Loops on all elements, directed edges from each vertex to each other vertex, in both directions.

Question 3

By looking at a matrix, how can you tell if the relation it represents is reflexive?

Answer 3

There are 1’s on the main diagonal.

Question 4

By looking at a matrix, how can you tell if the relation it represents is symmetric?

Answer 4

The (i,j)th entry is equal to the (j, i)th entry.

Question 5

By looking at a matrix, how can you tell if the relation it represents is transitive?

Answer 5

The (i,j)th entry of A * A is nonzero when the (i,j)th entry of A is nonzero.