GO Classes Set Theory And Algebra Practice Set 1 | Question: 4

Question

Let $A$ be the set of all ordered pairs of integers, that is, $A=Z \times Z$. Define a binary relation $R$ on $A$ as follows: for all $(a, b),(c, d) \in A$,
$$
(a, b) R(c, d) \Leftrightarrow a \leq c \text { and } b \leq d .
$$

• A. Is $R$ reflexive?
• B. Is $R$ symmetric?
• C. Is $R$ antisymmetric?
• D. Is $R$ transitive?
• E. Is $R$ an equivalence relation, a partial order, neither, or both?

Answer 1

→ R is Reflexive.

                             Let’s take one example for reflexive(a,b)<=>(1,1) and (c,d)<=> (2,2).

->R is not Symmetric.

                                   Let’s take (1,2) as (a,b). for symm there is also contain (2,1) but,it’s not satisfy with                                     condition. 

                                       so,it’s not symmetric.

->R is Antisymmetric.

                                   if it’s not symmetric so definitely antisymmetric.

->R is Transitive.

                            we take (1,2) and (2,3) as (a,b) and (b,c) respectively so,for transitive it has to be (a,c) it means                               (1,3) .and it’s satisfy with the condition.

                                                                                             so,it’s transitive .

 

                               and R is reflexive,antisymmetric and transitive .

                                                                                                            So,R is also a partial order relation.