(a) Let A = {0,1,2}. R = {(0,0), (0,1), (0,2), (1,1), (1,2), (2,2)} and S = {(0,0), (1,1), (2,2)} be two relations on A. (i) Show that R is a partial order relation. (ii) Is R a total order relation? (iii) Show that S is an equivalence relation. 1
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1) Since
Therefore is reflexive .
is antisymmetric if
Therefore is antisymmetric .
is transitive if for ,
Therefore is transitive.
Hence is a partial order relation.
2) is called total order relation if for any
As any two elements of are Related , therefore is total order.
3) As (0,0),(1,1),(2,2) , Therefore
is reflexive.
is symmetric if
Therefore is symmetric.
is transitive if
Clearly is transitive.
Therefore is a eqivalence relation.