retagged by
255 views
4 votes
4 votes

Let $\text{R}$ and $\text{S}$ be binary relations on a set $\text{A}.$ Suppose that $\text{R}$ is reflexive, symmetric, and transitive and that $\text{S}$ is symmetric, and transitive but is not reflexive.

Which statement is always true for any such $\text{R}$ and $\text{S}?$

  1. $\text{R} \cup \text{S}$ is symmetric but not reflexive and not transitive.
  2. $\text{R} \cup \text{S}$ is symmetric, reflexive, transitive
  3. $\text{R} \cup \text{S}$ is transitive and symmetric but not reflexive
  4. $\text{R} \cup \text{S}$ is reflexive and symmetric but may not be transitive.
retagged by

1 Answer

6 votes
6 votes
Since $\text{R}$ is reflexive so $\text{R} \cup \text{S}$ will also be reflexive. Since $\text{R, S}$ are symmetric, so $\text{R} \cup \text{S}$ also will be symmetric. But $\text{R} \cup \text{S}$ may not be transitive.

Consider this $: \text{A} = \{1,2,3\} , \text{R} = \{(1,1), (2,2), (3,3), (1,2),(2,1)\}$ and $\text{S} = \{(3,2), (2,3),(2,2),(3,3)\}$

$\text{R} \cup \text{S}$ is Not transitive.
edited by
Answer:

Related questions

3 votes
3 votes
1 answer
3