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For each relation, indicate whether the relation is a partial order, a strict order, or neither. If the relation is a partial or strict order, indicate whether the relation is also a total order. Justify your answers.(a)The domain is the set of all words in the English language (as defined by, say, Webster's dictionary). Word x is related to word y if x appears before y in alphabetical order. Assume that each word appears exactly once in the dictionary.

Respuesta :

a) The relation is not in total order as x"R"y or y"R"x may not happen.

b) The relation is of strict order but not of the total order.

a) The domain includes all runners in a race.

x is "R" y if x beats y

  • clearly, x "R" x implies no meaning and sense ⇒ irreflexive
  • If x "R" y ⇒ y does not beat x. Thus; asymmetric
  • If x "R' y and y "R" Z ⇒ Transitive.

Now, in a race; either x beats y or y beats x

So, x"R"y or y "R" x, but here at least two runners tied.

Thus, the relation is not in total order as x"R"y or y"R"x may not happen.

b) domain = Power set of S

x"R"y if |X| ≤ |Y|

  • clearly |X| ≤ |X|  ⇒ reflexive
  • If |X| ≤ |Y| and |Y| ≤ |X| ⇒ |X|=|Y| ⇒ Antisymmetric
  • If |X| ≤ |Y| and |Y| ≤ |Z| ⇒ |X|=|Z| ⇒ Transitive

c) S = {a,b,c,d}

The domain = Power set of S

x"R"y if |X| ≤ |Y|

  • clearly |X| < |X|  ⇒ Irreflexive
  • If |X| < |Y| and |Y| < |X| ⇒ |X|=|Y| ⇒ Antisymmetric
  • If |X| < |Y| and |Y| < |Z| ⇒ |X|<|Z| ⇒ Transitive
  • Thus, the relation is of strict order but not of the total order.

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