Properties of Relations

I'm having real trouble being able to identify when a relation is reflexive, irreflexive, symmetric, antisymmetric, and transitive.

For instance, here is a relation on the set {1, 2, 3, 4, 5}

R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}

What exactly do you look for when determining if the set has each quality mentioned above? I know the basics (i.e. reflexive is for all a, aRa) but I seem to be getting these wrong each time. If someone could map out how they approach this problem in their mind I would really appreciate it.

Quote:

Originally Posted by brand_182

I'm having real trouble being able to identify when a relation is reflexive, irreflexive, symmetric, antisymmetric, and transitive.

For instance, here is a relation on the set {1, 2, 3, 4, 5}

R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}

What exactly do you look for when determining if the set has each quality mentioned above? I know the basics (i.e. reflexive is for all a, aRa) but I seem to be getting these wrong each time. If someone could map out how they approach this problem in their mind I would really appreciate it.

Here, use $_aR_b$ to mean $(a,b) \in R$ .

The relation R is reflexive if $(a,a) \in R$ for all $a \in \{ 1,2,3,4,5 \}$

The relation is irreflexive if $(a,a) \not \in R$ for all $a \in \{ 1,2,3,4,5 \}$

The relation is symmetric if $(a,b) \in R$ implies $(b,a) \in R$ , for all $a,b \in \{ 1,2,3,4,5 \}$

The relation is antisymmetric if $(a,b) \in R$ and $(b,a) \in R$ implies $a = b$ , for all $a,b \in \{ 1,2,3,4,5 \}$

The relation is transitive if $(a,b) \in R$ and $(b,c) \in R$ implies $(a,c) \in R$ , for all $a,b,c \in \{ 1,2,3,4,5 \}$

The important word is "all," you must check that these definitions hold for all elements in R

thank you very much!