Question

Let A be {a, b, c}. Let the relation R be {(c, b), (a, a), (b, c)}. Which of the following statements about R is true?
a. R is not reflexive, is not symmetric, and is not transitive.

b. R is reflexive, is symmetric, and is not transitive.

c. R is reflexive, is not symmetric, and is not transitive.

d. R is not reflexive, is symmetric, and is not transitive.

Answer 1

Let $A= \{a, b, c\}$. Let the relation $R= \{(c, b), (a, a), (b, c)\}$. 

Since $(b,b)\notin R,$ we conclude that the relation $R$ is not reflexive. Taking into account that $(b,c)\in R$ and $(c,b)\in R,$ but $(b,b)\notin R,$ we conclude that $R$ is not transitive. Since $(b,c)\in R$ implies $(c,b)\in R$ and $(a,a)\in R$ implies $(a,a)\in R,$ the relation $R$ is symmetric.

Answer: d