In each case below, a relation on the set {1, 2, 3} is given. Of the three properties, reflexivity, symmetry, and transitivity, determine which ones the relation has. Give reasons.

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.