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 S, the transitive closure of S. S; =

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Chapter2: Second-order Linear Odes
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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 \( S_t \), the transitive closure of \( S \).

\[ S_t = \]

Explanation:

- The relation \( S \) consists of a set of ordered pairs derived from the set \( A = \{0, 1, 2, 3\} \).
- The transitive closure \( S_t \) is the smallest transitive relation on \( A \) that contains \( S \). It includes all the pairs in \( S \) and any additional pairs needed to satisfy transitivity. 

You may solve for \( S_t \) by applying the transitive property: if a relation \( (a, b) \) and \( (b, c) \) exist, then \( (a, c) \) must also be included.
Transcribed Image Text: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 \( S_t \), the transitive closure of \( S \). \[ S_t = \] Explanation: - The relation \( S \) consists of a set of ordered pairs derived from the set \( A = \{0, 1, 2, 3\} \). - The transitive closure \( S_t \) is the smallest transitive relation on \( A \) that contains \( S \). It includes all the pairs in \( S \) and any additional pairs needed to satisfy transitivity. You may solve for \( S_t \) by applying the transitive property: if a relation \( (a, b) \) and \( (b, c) \) exist, then \( (a, c) \) must also be included.
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