Which ordered pairs need to be added to the relationp = {(a,a),(a,d),(b,b),(b,d),(c,c),(d,a),(d,b)}on the set X = {a,b,c,d} to create the equivalence relation p* generated by p?
Question
Which ordered pairs need to be added to the relationp = {(a,a),(a,d),(b,b),(b,d),(c,c),(d,a),(d,b)}on the set X = {a,b,c,d} to create the equivalence relation p* generated by p?
Solution
To generate the equivalence relation p* from the relation p, we need to ensure that the relation is reflexive, symmetric, and transitive.
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Reflexivity: For a relation to be reflexive, every element in the set X must be related to itself. In the given relation p, (a,a), (b,b), and (c,c) are already present. However, (d,d) is missing. So, we need to add (d,d) to the relation.
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Symmetry: For a relation to be symmetric, if (a,b) is in the relation, then (b,a) must also be in the relation. In the given relation p, (a,d), (b,d), and (d,a), (d,b) are present. However, (d,a) and (d,b) are missing. So, we need to add (d,a) and (d,b) to the relation.
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Transitivity: 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. In the given relation p, we have (a,d) and (d,b), so we need to add (a,b). Similarly, we have (b,d) and (d,a), so we need to add (b,a).
So, the ordered pairs that need to be added to the relation p to create the equivalence relation p* are (d,d), (d,a), (d,b), (a,b), and (b,a).
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