Answered

From health tips to tech hacks, find it all on IDNLearn.com. Ask your questions and receive reliable and comprehensive answers from our dedicated community of professionals.

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|.

Sagot :

Ajeigbeibraheemidn

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.

Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Your search for answers ends at IDNLearn.com. Thanks for visiting, and we look forward to helping you again soon.