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Identifying partial, strict, and total orders.
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.
(i) The domain is the set of all runners in a race. x is related to y if x beat y in the race. At least two runners in the race tied.
(j) S = {a, b, c, d}. The domain is P(S), the power set of S. For X, Y that are subsets of S, X is related to Y if |X| ≤ |Y|.
(k) S = {a, b, c, d}. The domain is P(S), the power set of S. For X, Y that are subsets of S, X is related to Y if |X| < |Y|.

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Answer:

Step-by-step explanation:

From the given information:

(i)

The domain includes all runners in a race.

x is "R" y if x beats y

  1. clearly, x "R" x implies no meaning and sense ⇒ irreflexive
  2. If x "R" y ⇒ y does not beat x. Thus; asymmetric
  3. 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.

(j)

S = {a,b,c,d}

The domain = Power set of S

x"R"y if |X| ≤ |Y|

  1. clearly |X| ≤ |X|  ⇒ reflexive
  2. [tex]If \ |X| \le |Y| \ and \ |Y| \le |X|[/tex] ⇒ |X|=|Y| ⇒ Antisymmetric
  3. [tex]If |X| \ \le \ |Y| \ and \ |Y| \ \le \ |Z|[/tex] ⇒ |X| ≤ |Z| ⇒ Transitive

Thus, the relation is a partial order.

(k)

S = {a,b,c,d}

The domain = Power set of S

x"R"y if |X| ≤ |Y|

  1. clearly |X| < |X|  ⇒ Irreflexive
  2. [tex]\text{If } |X| < |Y| \ and \ |Y| < |X|} \implies Antisymmetric[/tex]
  3. [tex]|X| < |Y| \ and \ |Y| < |Z| \implies |X| < |Z|[/tex] ⇒ Transitive
  4. Thus, the relation is of strict order but not of the total order.

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