Question

Construct a relation on the set {a, b, c, d} that is

a. reflexive, symmetric, but not transitive.

b. irreflexive, symmetric, and transitive.

c. irreflexive, antisymmetric, and not transitive.

d. reflexive, neither symmetric nor antisymmetric, and transitive.

Answer 1

a. $R = \{ (a,a),\,(b,b),\,(c,c),\,(d,d),\,(a,b),\,(b,a)\,(a,c),\,(c,a),\,\,(b,c),(c,b),\,(b,d),\,(d,b)\}$

For any $x \in \{ a,b,c,d\}$ $(x,x) \in R$ , so, R is reflexive.

For any $x,y \in \{ a,b,c,d\}$ if $(x,y) \in R$ then $(y,x) \in R$ , so, R is symmetric.

$(a,b) \in R,\,(b,d) \in R$ , but $(a,d) \notin R$ , so, R is not transitive.

b. Let R is this relation.

Since R is symmetric then for any $x,y \in \{ a,b,c,d\}$ if $(x,y) \in R$ then $(y,x) \in R$.

But then (since R is transitive) $(x,x)∈R$ . But then R isn't irreflexive, which contradicts the condition, therefore, such a relation does not exist.

c. $R = \{ (a,b),\,(a,c),\,(b,d)\}$

For any $x \in \{ a,b,c,d\}$ $(x,x) \notin R$ , so, R is irreflexive.

For any $x,y \in \{ a,b,c,d\}$ if $(x,y) \in R$ then $(y,x) \notin R$ , so, R is antisymmetric.

$(a,b) \in R,\,(b,d) \in R$ , but $(a,d) \notin R$ , so, R is not transitive.

d. $R = \{ (a,a),\,(b,b),\,(c,c),\,(d,d),\,(a,b),(b,a),\,(a,c)\}$

For any $x \in \{ a,b,c,d\}$ $(x,x) \in R$ , so, R is reflexive.

$(a,b) \in R,\,(b,a) \in R$ , so R isn't antisymmetric, $(a,c) \in R,\,(c,a) \notin R$ so R isn't symmetric.

for any $x,y,z \in \{ a,b,c,d\}$ $(x,y) \in R \wedge (y,z) \in R \Rightarrow (x,z) \in R$ , so, R is transitive.