Answer:
a. is symmetric but not reflexive and transitive
b. is reflexive and transitive but not symmetric
c. is reflexive, symmetric and transitive
Step-by-step explanation:
The cases are missing in the question.
Let the cases be as follows:
a. R = {(1, 3), (3, 1), (2, 2)}
b. R = {(1, 1), (2, 2), (3, 3), (1, 2)}
c. R = ∅
R is defined on the set {1, 2, 3}
- R is reflexive if for all x in {1, 2, 3} xRx
- R is symmetric if for all x,y in {1, 2, 3} if xRy then yRx
- R is transitive if for all x,y,z in {1, 2, 3} if xRy and yRz then xRz
a. R = {(1, 3), (3, 1), (2, 2)} is
- not reflexive since for x=1, (1,1) is not in R
- symmetric since for all x,y in {1, 2, 3} if xRy then yRx
- not transitive because (1, 3), (3, 1) is in R but (1,1) is not.
b. R = {(1, 1), (2, 2), (3, 3), (1, 2)} is
is reflexive because (1, 1), (2, 2), (3, 3) is in R
is not symmetric because for (1,2) (2,1) is not in R
is transitive becaue for (1,1) and (1,2) we have (1,2) in R
c. R = ∅ is
reflexive, symmetric and transitive because it satisfies the definitions since there is no counter example.
