Transitive Relations

Tell whether the following is transitive.

R is:
{(5,3),(4,5),(4,4),(4,2),(4,3),(2,3),(1,5), (1,2), (1,3), (1,1)}

To determine whether the given relation R is transitive, we need to check if for every pair of tuples (a, b) and (b, c) in R, the pair (a, c) is also in R.

Let's go through each pair (a, b) and (b, c) in R and check if the pair (a, c) is also in R:

1. (5, 3) and (3, x): There is no tuple in R with 3 as the first element, so we cannot determine if (5, x) is in R.

2. (4, 5) and (5, x): There is no tuple in R with 5 as the first element, so we cannot determine if (4, x) is in R.

3. (4, 4) and (4, x): There are tuples in R with 4 as the first element: (4, 5), (4, 2), (4, 3). In all cases, (4, x) is not in R.

4. (4, 2) and (2, x): There is a tuple in R with 2 as the first element: (2, 3). So, (4, 3) is in R.

5. (4, 3) and (3, x): There is no tuple in R with 3 as the first element, so we cannot determine if (4, x) is in R.

6. (2, 3) and (3, x): There is no tuple in R with 3 as the first element, so we cannot determine if (2, x) is in R.

7. (1, 5) and (5, x): There is no tuple in R with 5 as the first element, so we cannot determine if (1, x) is in R.

8. (1, 2) and (2, x): There is a tuple in R with 2 as the first element: (2, 3). So, (1, 3) is in R.

9. (1, 3) and (3, x): There is no tuple in R with 3 as the first element, so we cannot determine if (1, x) is in R.

10. (1, 1) and (1, x): There are tuples in R with 1 as the first element: (1, 5), (1, 2), (1, 3). In all cases, (1, x) is not in R.

Since there are pairs (a, b) and (b, c) in R where (a, c) is not in R, the relation R is not transitive.