Binary relations: Can someone see if I have done this correctly?

Question

I would appreciate some help.

Here are the binary relations:

Below we are defining some binary relations over the set {a, b, c}

1. S ₁: {〈a, a⟩, 〈a, b⟩, 〈b, b⟩, 〈b, c⟩}
2. S ₂: {〈a, a⟩, 〈a, b⟩, 〈b, b⟩}
3. S ₃: {〈a, a⟩, 〈a, b⟩, 〈b, b⟩, 〈b, c⟩, }
4. S ₄: {〈a, a⟩, 〈a, b⟩, 〈b, a⟩, 〈b, b⟩, 〈c, c⟩}
5. S ₅: {〈a, a⟩, 〈b, c⟩, 〈c, b⟩}

Here is the questions:

1. Which element(s) is missing in S ₁ so it can become a reflexive relation?
2. Which element(s) is missing in S ₂ so it can become a symmetric relation?
3. Which element(s) is missing in S ₃ so it can become a transitive relation?
4. Which element(s) can you remove from S ₄ so it can become anti-symmetric relation?
5. Which element(s) can you remove from S ₅ so it can become a irreflexive relation?
6. Which of the binary relations above are reflexive?
7. Which of the binary relations above are symmetric?
8. Which of the binary relations above are transitive?

Here is my answers:

1. We can insert 〈c, c⟩ in S ₁ and it will become a reflexive relation.

2. We can insert 〈b, c⟩, 〈c, c⟩, 〈a, c⟩, 〈c, a⟩, 〈b, a⟩, 〈c, b⟩ in S ₂ and it will become a symmetric relation

3. We can insert 〈a, c⟩, 〈c, c⟩ in S ₃ and it will become a transitive relation

4. We can take out 〈b, a⟩ from S ₄ and it will become a anti-symmetric relation

5. We can take out 〈a, a⟩ from S ₅ and it will become a irreflexive relation

6. S ₂ is a reflexive relation

7. S ₄ and S ₅ are symmetric relations

8. S ₂ and S ₄ are transitive relations

As you can see, i've figured out all the answers for the tasks, but i'm unsure if the answers are correct. I would appreciate if someone could see if I have done anything wrong here or if there is something i'm missing.

Thanks alot!

Answer 1

In order to check if a relation is reflexive you need to know on which set the relation is defined. My guess would be that you are meant to use $\{a,b,c\}$ in all cases. This assumption makes $S_2$ not a reflexive relation, because it is missing $(c,c)$.

Your second answer is correct, but just inserting $(b,a)$ is sufficient.

Your third answer is correct, but just inserting $(b,c)$ is sufficient.

Your sixth answer is wrong, only $S_4$ is reflexive under the assumption mentioned earlier.

The other answers are correct.