Let’s refresh the definitions that are relevant to this exercise.
Definition: “A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.”
Definition: “A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a,b) ∈ R, for all a,b ∈ A.
Definition: “A relation R on a set A is called transitive if whenever (a,b)∈R and (b,c)∈R, then (a,c) ∈ R, for all a,b,c ∈ A.”
The definitions above are from the textbook “Discrete Mathematics and its Applications” by Rosen.
Now, let’s solve the exercise.
Let a∈S, (a, a)∉R because of R=∅. Therefore, R is not reflexive.
As per the definition of symmetric relation, if (b, a) ∈ R whenever (a, b) ∈ R, for all a, b ∈ A, then the relation is symmetric. This holds for R==∅. Therefore, R is symmetric.
Similarly, As per the definition of transitive relation, whenever (a,b)∈R and (b,c)∈R, then (a,c) ∈ R, for all a,b,c ∈ A. This condition also holds for R=∅. Therefore, R is transitive.
If this is confusing for you, try to find an ordered pair (x, y) that belongs to R, and then prove that (y, x) does not belong to R.
Related exercises:
- Give an example of a relation on a set that is a) both symmetric and antisymmetric. b) neither symmetric nor antisymmetric
- Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x,y)∈R if and only if
- a) List all the ordered pairs in the relation R = {(a,b) | a divides b} on the set {1,2,3,4,5,6}. b) Display this relation graphically, as was done in Example 4. c)Display this relation in tabular form, as was done in Example 4
- Show that the relation R=∅ on the empty set S=∅ is reflexive, symmetric, and transitive
- Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if
