For each relation below, determine if they are reflexive, symmetric, anti-symmetric, and transitive. (a) X= { 1, 2, 3, 4} R1={(1, 2),(2, 3),(3, 4)} (b) X = {a, b, c, d, e} R1 = { (a, a), (a, b), (a, e), (b, b), (b, e), (c, c), (c, d), (d, d), (e, e) } (c) X= { 1, 2, 3, 4} R1= {(1, 3),(1, 4),(2, 3),(2, 4),(3, 1),(3, 4)}

A relation $R$ on a set $A$ is called reflexive if $(a, a) ∈ R$ for every element $a ∈ A.$

A relation $R$ on a set $A$ is called symmetric if $(b, a) ∈ R$ whenever $(a, b) ∈ R,$ for all $a, b ∈ A.$

A relation $R$ on a set $A$ such that for all $a, b ∈ A,$ if $(a, b) ∈ R$ and $(b, a) ∈ R,$ then $a = b$ is

called antisymmetric.

A relation $R$ on a set $A$ is called transitive if whenever $(a, b) ∈ R$ and $(b, c) ∈ R,$ then $(a, c) ∈ R,$ for all $a, b, c ∈ A.$

(a)

$X= \{ 1, 2, 3, 4\}$

$R1=\{(1, 2),(2, 3),(3, 4)\}$

Not reflexive because we do not have $(1, 1),(2,2), (3, 3),$ and $(4, 4).$

Not symmetric because while we have $(1, 2),$ we do not have $(2, 1).$

Antisymmetric.

Not transitive because we do not have $(1, 3)$ for $(1, 2)$ and $(2, 3).$  

(b)

$X= \{ a, b, c, d, e\}$

$R1=\{(a, a), (a, b), (a, e), (b, b), (b, e), (c, c), (c, d),$

$(d, d), (e, e) \}$

Reflexive because we have $(a, a), (b, b),(c, c), (d, d),$ and $(e,e).$

Not symmetric because while we have $(a, b),$ we do not have $(b, a).$

Antisymmetric.

Transitive.  

(c)

$X= \{ 1, 2, 3, 4\}$

$R1=\{(1, 3),(1, 4),(2, 3),(2, 4),(3, 1),(3, 4)\}$

Not reflexive because we do not have $(1, 1),(2,2), (3, 3),$ and $(4, 4).$

Not symmetric because while we have $(1, 4),$ we do not have $(4, 1).$

Not antisymmetric because we have both $(1,3)$ and $(3, 1).$

Not transitive because we do not have $(1, 1)$ for $(1, 3)$ and $(3, 1).$