Question

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.

Answer 1

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.