Question

let a relation s be defined on a = {0, 1, 2, 3} as follows. s = (0, 0), (0, 3), (1, 0), (1, 2), (2, 0), (3, 2) find st, the transitive closure of s.

Answer 1

To find ST, the transitive closure of S, we check for pairs (a, c) that need to be added to S based on existing pairs (a, b) and (b, c), resulting in ST being {(0, 0), (0, 2), (0, 3), (1, 0), (1, 2), (2, 0), (3, 2)}.

Step-by-step explanation:

To find ST, the transitive closure of the relation S, we need to add the minimal number of pairs to S such that for any three elements a, b, and c in set A, if the pairs (a, b) and (b, c) are in S, then the pair (a, c) must also be in S for the relation to be considered transitive.

Original Relation S:
{(0, 0), (0, 3), (1, 0), (1, 2), (2, 0), (3, 2)}

To achieve transitivity, we examine the pairs. We see that we have (0, 3) and (3, 2), so we must include the pair (0, 2). Similarly, from (1, 2) and (2, 0), we must include (1, 0), but it is already there. No other pairs need to be added for transitivity. Therefore, the transitive closure ST is:
{(0, 0), (0, 2), (0, 3), (1, 0), (1, 2), (2, 0), (3, 2)}

Answer 2

The transitive closure of the relation S = {(0, 0), (0, 3), (1, 0), (1, 2), (2, 0), (3, 2)} on the set A = {0, 1, 2, 3} is St = {(0, 0), (0, 3), (0, 2), (1, 0), (1, 2), (2, 0), (3, 2)}.

The transitive closure of a relation is a set of ordered pairs that includes all pairs in the original relation, as well as any pairs that can be formed by combining pairs from the original relation.

To find the transitive closure of the relation S = {(0, 0), (0, 3), (1, 0), (1, 2), (2, 0), (3, 2)} on the set A = {0, 1, 2, 3}, we need to check for any pairs that can be formed by combining pairs from the original relation.

In this case, we can see that if (0, 3) and (3, 2) are both in the relation, then (0, 2) should also be included in the transitive closure.

Similarly, if (1, 2) and (2, 0) are both in the relation, then (1, 0) should also be included.

Applying this logic, we find that the transitive closure of S is St = {(0, 0), (0, 3), (0, 2), (1, 0), (1, 2), (2, 0), (3, 2)}.