Answered

Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

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 :

ajeigbeibraheem

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.

We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.