Answer to Question #198877 in Discrete Mathematics for Vina

Question

Suppose U = {1, 2, 3, a, b, c} is a universal set with the subset A = {a, c, 2, 3}. Let R = { (a, a), (a, c), (3, c), (3, a), (2, 3), (2, a), (2, c), (c, 2) } be a relation on A. Answer questions 11 & 12 by using the given sets A, U and the relation R.

Question 11

Which one of the following statements regarding relation R is true? 1. R is a weak total order. 2. R is a strict total order. 3. R is a weak partial order. 4. R is not an equivalence relation.  Question 12 Which ordered pairs should be removed from relation R in order for the changed relation R1 (say) to be a strict partial order? 1. only (2, c) 2. only (a, a) 3. (a, a) & (c, 2) 4. (a, a) & (2, c) 

Answer 1

Question 11

Reflexivity is not for all elements of subset A. For example, relation R has not pair $(c.c)$

Answer: 4. R is not an equivalence relation.

Question 12

A strict partial order is transitive, antisymmetric (no ordered pair and its reverse in the relation), and irreflexive ( a binary relation R on a set A is called irreflexive if aRa does not hold for any a∈A).

So, (a, a) & (2, c)  should be removed from R.

Answer: 4. (a, a) & (2, c)