2 votes 2 votes Which of the following is/are correct? A relation that is reflexive, anti-symmetric, and transitive is called a partial order. When every two elements in a set are comparable, the relation is called a total ordering (linear ordering). The poset $(\mathbb{Z},\leq)$ is not totally ordered, where $\mathbb{Z}$ is the set of integers. The inclusion relation $\subseteq$ is a partial ordering on the power set of a set. I and II only I and III only I, II and III only I, II and IV only Set Theory & Algebra go2025-dm-3 partial-order + – gatecse asked Jul 5, 2020 gatecse 146 views answer comment Share Follow See 1 comment See all 1 1 comment reply shadymademe commented Nov 10, 2023 reply Follow Share @gatecse SIr, the definition in option II is incomplete, even an equivalence relation on a set of size $>=2$ will satisfy the condition of every two elements being comparable but that equivalence relation is not anti-symmetric, and hence not even a partial order relation (total order relation). 0 votes 0 votes Please log in or register to add a comment.
Best answer 2 votes 2 votes The poset $(\mathbb{Z}, \leq)$ is totally ordered, because for any integers $a,b, a\leq b$ or $b\leq a.$ Other three statements are true. So, the correct answer is $(D)$. gatecse answered Jul 5, 2020 • selected Jul 2, 2022 by Arjun gatecse comment Share Follow See all 2 Comments See all 2 2 Comments reply Harshit Dubey commented Jul 16, 2023 reply Follow Share Can someone please clarify the D part @Deepak Poonia sir 0 votes 0 votes Deepak Poonia commented Jul 17, 2023 reply Follow Share @Harshit DubeyFor any set $S,$ $(P(S), \subseteq )$ is a POSET as well as Distributive Complemented Lattice.For any set $S,$ $(P(S), \subseteq )$ is a boolean algebra(boolean lattice). 2 votes 2 votes Please log in or register to add a comment.