Answered

Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A 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.

Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.