Discrete Mathmatics Computer Science List the $16$ different relations on the set $\{0,1\}.$

Which of the $16$ relations on $\{0,1\},$ which you listed in Exercise $42,$ are

a$)$ reflexive?

b$)$ irreflexive?

c$)$ symmetric?

d$)$ antisymmetric?

e$)$ asymmetric?

f$)$ transitive?

DEFINITION $1$

Let A and $B$ be sets. A binary relation from $A$ to $B$ is a subset of $AB.$

DEFINITION $2$

A relation on a set $A$ is a relation from $A$ to $A.$

DEFINITION $3$

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

DEFINITION $4$

A relation $R$ on a set $A$ is called symmetric if $(b,a)inR$ whenever $(a,b)inR,$ for all $a,$binA. A relation $R$ on a set A such that for all $a,$binA, if $(a,b)inR$ and $(b,a)inR,$ then $a=b$ is called antisvmmetric.

EFINITION $5$

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

DEFINITION $6$

Let $R$ be a relation from a set $A$ to a set $B$ and $S$ a relation from $B$ to a set $C.$ The composite of $R$ and $S$ is the relation consisting of ordered pairs $(a,c),$ where ainA,cinC, and for which there exists an element binB such that $(a,b)inR$ and $(b,c)inS.$ We denote the composite of $R$ and $S$ by $S$@$R.$

JEFINITION $7$

Let $R$ be a relation on the set $A.$ The powers $R^n,n=1,2,3,$dots, are defined recursively by $R^1=R,$ and $,R^{n+1}=R^n$@$R.$

THEOREM $1$

The relation $R$ on a set $A$ is transitive if and only if $R^n$subeR for $n=1,2,3,$dots