Relation Math Question

Question

Let R1​={(a,1),(b,1),(c,1)} be a relation on the set {1,2,3,4,5,a,b,c}, and let R2​={(1,2),(2,3),(1,3),(1,1)} be a relation on the set {1,2,3}.

Which of the following statements are correct? Tick all the corresponding answers.

a) Both R1​ and R2​ are reflexive.

b) R1​ is transitive.

c) R2​ is transitive.

d) R1​ is symmetric.

e) R2​ is symmetric.

Answer 1

a) Incorrect: (1,1) is not true for R1

b) Correct? I'm not sure about this. I believe it is "vacuously true" because the element (1,*) is false, hence (x,1) AND (1,*) => (x,*), is technically true for all x=a,b,c because the antecedent is false.

c) Correct? Again, the only one I'm concerned about is (1,3) because (3,*) is false. However, for the same reason as above, I believe that this is correct

d) Incorrect. (a,1) is in R1, but (1,a) is not

e) Incorrect. (1,2) is in R2, but (2,1) is not