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.

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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.

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

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

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

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

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

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


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

thank you very much!