Which ordered pairs need to be added to the relationp = {(a,a),(a,c),(b,b),(b,d),(c,a),(c,c),(d,b),(d,d)}on the set X = {a,b,c,d} to create the equivalence relation p* generated by p? A (a,a) B (a,b) C (a,c) D (a,d) E (b,a) F (b,b) G (b,c) H (b,d) I (c,a) J (c,b) K (c,c) L (c,d) M (d,a) N (d,b) O (d,c) P (d,d) Q None of them
Question
Which ordered pairs need to be added to the relationp = {(a,a),(a,c),(b,b),(b,d),(c,a),(c,c),(d,b),(d,d)}on the set X = {a,b,c,d} to create the equivalence relation p* generated by p? A (a,a) B (a,b) C (a,c) D (a,d) E (b,a) F (b,b) G (b,c) H (b,d) I (c,a) J (c,b) K (c,c) L (c,d) M (d,a) N (d,b) O (d,c) P (d,d) Q None of them
Solution
To generate an equivalence relation from a given relation, the relation must be reflexive, symmetric, and transitive.
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Reflexivity: Every element must be related to itself. In the set X = {a,b,c,d}, all elements are already related to themselves in the relation p. So, we don't need to add any pairs for reflexivity.
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Symmetry: If (a,b) is in the relation, then (b,a) must also be in the relation. Looking at the relation p, we see that (a,c), (b,d), (c,a), and (d,b) are in the relation, but their symmetric pairs (c,a), (d,b), (a,c), and (b,d) are also in the relation. So, we don't need to add any pairs for symmetry.
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Transitivity: If (a,b) and (b,c) are in the relation, then (a,c) must also be in the relation. Looking at the relation p, we see that (a,c) and (c,a) are in the relation, but (a,a) is also in the relation. Similarly, (b,d) and (d,b) are in the relation, but (b,b) is also in the relation. So, we don't need to add any pairs for transitivity.
Therefore, the answer is Q. None of them.
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