Thread: Relations with ø

Let A=ø, and consider the empty relation, ø, on A. Which properties does ø satisfy?

Is it reflexive, irreflexive, symmetric, antisymmetric, transitive?

Originally Posted by kturf

Let A=ø, and consider the empty relation, ø, on A. Which properties does ø satisfy?

Is it reflexive, irreflexive, symmetric, antisymmetric, transitive?

Use the fact that if p is false, then the statement "if p then q" is true no matter what q is.

"reflexive" means "if x is in A, then (x,x) is in the relation". But, since A is empty, "x is in A" is false for all x.

"irreflexive" means "if x is in A, then (x,x) is NOT in the relation". But, since A is empty, "x is in A" is false for all x.

"Symmetric' means "if (x,y) is in the relation then (y,x) is in the relation". But, since ø is the empty relation, "(x,y) is in the relation" is false for all x, y.

"Anti-symmetric" means "if (x,y) is in the relation then (y,x) is NOT in the relation. But, since ø is the empty relation "(x,y) is in the relation" is false for all x,y.

"transitive" means "if (x,y) and (y,z) are in the relation then (x,z) is in the relation". But, since ø is the empty relation, "(x,y) and (y,z) are in the relation" is false for all x,y,z.

Once again, if the hypothesis, p, of the statement "if p then q" is false the statement itself is true.