Question

Define the following relations on A={4,6,7,9}. Answer each of the following questions by listing the index of the relation. For example, for Part 1), if R1 and R2 are reflexive, simply enter 1, 2 as the answer.

R0={(4,4),(9,9),(9,6),(7,7),(6,7),(9,4)}

R1={(7,7),(4,4),(6,6),(9,9)}

1) Which relations are reflexive?

2) Which relations are symmetric?

3) Which relations are anti-symmetric?

4) Which relations are transitive?

R2={(4,4),(6,4),(9,9),(7,7),(6,6),(4,6)}

R3={(7,7),(4,9),(7,6),(6,9),(9,4),(6,7),(9,6)}

R4={(7,6),(7,9),(7,7),(4,6),(4,9),(7,4)}

Answer 1

1) R1 and R2 are reflexive. 2) None of the given relations are symmetric. 3) None of the given relations are anti-symmetric. 4) None of the given relations are transitive.

Step-by-step explanation:

1) The reflexive relations are R1 and R2. R1 is reflexive because it contains the pairs (7,7), (4,4), (6,6), and (9,9), which include every element in A paired with itself. R2 is also reflexive because it contains the pairs (4,4), (6,6), and (9,9), which include every element in A paired with itself.

2) None of the given relations are symmetric. For a relation to be symmetric, if (a,b) is in the relation, then (b,a) must also be in the relation. None of the given relations have this property.

3) None of the given relations are anti-symmetric. For a relation to be anti-symmetric, if (a,b) is in the relation and (b,a) is also in the relation, then a must be equal to b. None of the given relations have this property.

4) None of the given relations are transitive. For a relation to be transitive, if (a,b) and (b,c) are in the relation, then (a,c) must also be in the relation. None of the given relations have this property.