Let A={0,1,2}. R={(0,0),(0,1),(0,2),(1,1),(1,2),(2,2)} and S={(0,0),(1,1),(2,2)} be two relations on A. 1.Show that R is a partial order relation 2.Is R a total order relation? 3.Show that S is an equivalence relation.

1.1 Since $\{(0,0),(1,1),(2,2)\}\subset R$ , then $R$ is reflexive.

1.2 For each pair $(a,b)\in R$ if $(b,a)\in R$ then $a=b$ , hence $R$ is a symmetric relation.

1.3 If $(a,b)\in R$ and $(b,c)\in R$ , then $(a,c)\in R$ , hence $R$ is transitive.

By 1.1, 1.2 and 1.3 $R$ is a partial order relation.

It is easy to see that $R$ is natural order $(\leq)$ relation on the set $A=\{0,1,2\}$, hence (1) $R$ is a partial order relation and $R$ is total.

2. $R$ is total.

$(a,b)\in S$ if and only if $a=b$ , hence

3.1 Since $\{(0,0),(1,1),(2,2)\}\subset S$, then $S$ is reflexive.

3.2 If $(a,b)\in S$ , then $a=b$ and $(b,a)\in S$ , hence $S$ is a symmetric relation.

3.3 If $(a,b)\in S$ and $(b,c)\in S$, then $a=b=c$ and $(a,c)\in S$ , hence $S$ is transitive.

By 3.1, 3.2 and 3.3 $S$ is an equivalence relation.

1.1 Since $\{(0,0),(1,1),(2,2)\}\subset R$ , then $R$ is reflexive.

1.2 For each pair $(a,b)\in R$ if $(b,a)\in R$ then $a=b$ , hence $R$ is a symmetric relation.

1.3 If $(a,b)\in R$ and $(b,c)\in R$ , then $(a,c)\in R$ , hence $R$ is transitive.

By 1.1, 1.2 and 1.3 $R$ is a partial order relation.

It is easy to see that $R$ is natural order $(\leq)$ relation on the set $A=\{0,1,2\}$, hence (1) $R$ is a partial order relation and $R$ is total.

2. $R$ is total.

$(a,b)\in S$ if and only if $a=b$ , hence

3.1 Since $\{(0,0),(1,1),(2,2)\}\subset S$, then $S$ is reflexive.

3.2 If $(a,b)\in S$ , then $a=b$ and $(b,a)\in S$ , hence $S$ is a symmetric relation.

3.3 If $(a,b)\in S$ and $(b,c)\in S$, then $a=b=c$ and $(a,c)\in S$ , hence $S$ is transitive.

By 3.1, 3.2 and 3.3 $S$ is an equivalence relation.

1.1 Since $\{(0,0),(1,1),(2,2)\}\subset R$ , then $R$ is reflexive.

1.2 For each pair $(a,b)\in R$ if $(b,a)\in R$ then $a=b$ , hence $R$ is a symmetric relation.

1.3 If $(a,b)\in R$ and $(b,c)\in R$ , then $(a,c)\in R$ , hence $R$ is transitive.

By 1.1, 1.2 and 1.3 $R$ is a partial order relation.

It is easy to see that $R$ is natural order $(\leq)$ relation on the set $A=\{0,1,2\}$, hence (1) $R$ is a partial order relation and $R$ is total.

2. $R$ is total.

$(a,b)\in S$ if and only if $a=b$ , hence

3.1 Since $\{(0,0),(1,1),(2,2)\}\subset S$, then $S$ is reflexive.

3.2 If $(a,b)\in S$ , then $a=b$ and $(b,a)\in S$ , hence $S$ is a symmetric relation.

3.3 If $(a,b)\in S$ and $(b,c)\in S$, then $a=b=c$ and $(a,c)\in S$ , hence $S$ is transitive.

By 3.1, 3.2 and 3.3 $S$ is an equivalence relation.

1.1 Since $\{(0,0),(1,1),(2,2)\}\subset R$ , then $R$ is reflexive.

1.2 For each pair $(a,b)\in R$ if $(b,a)\in R$ then $a=b$ , hence $R$ is a symmetric relation.

1.3 If $(a,b)\in R$ and $(b,c)\in R$ , then $(a,c)\in R$ , hence $R$ is transitive.

By 1.1, 1.2 and 1.3 $R$ is a partial order relation.

It is easy to see that $R$ is natural order $(\leq)$ relation on the set $A=\{0,1,2\}$, hence (1) $R$ is a partial order relation and $R$ is total.

2. $R$ is total.

$(a,b)\in S$ if and only if $a=b$ , hence

3.1 Since $\{(0,0),(1,1),(2,2)\}\subset S$, then $S$ is reflexive.

3.2 If $(a,b)\in S$ , then $a=b$ and $(b,a)\in S$ , hence $S$ is a symmetric relation.

3.3 If $(a,b)\in S$ and $(b,c)\in S$, then $a=b=c$ and $(a,c)\in S$ , hence $S$ is transitive.

By 3.1, 3.2 and 3.3 $S$ is an equivalence relation.

1.1 Since $\{(0,0),(1,1),(2,2)\}\subset R$ , then $R$ is reflexive.

1.2 For each pair $(a,b)\in R$ if $(b,a)\in R$ then $a=b$ , hence $R$ is a symmetric relation.

1.3 If $(a,b)\in R$ and $(b,c)\in R$ , then $(a,c)\in R$ , hence $R$ is transitive.

By 1.1, 1.2 and 1.3 $R$ is a partial order relation.

It is easy to see that $R$ is natural order $(\leq)$ relation on the set $A=\{0,1,2\}$, hence (1) $R$ is a partial order relation and $R$ is total.

2. $R$ is total.

$(a,b)\in S$ if and only if $a=b$ , hence

3.1 Since $\{(0,0),(1,1),(2,2)\}\subset S$, then $S$ is reflexive.

3.2 If $(a,b)\in S$ , then $a=b$ and $(b,a)\in S$ , hence $S$ is a symmetric relation.

3.3 If $(a,b)\in S$ and $(b,c)\in S$, then $a=b=c$ and $(a,c)\in S$ , hence $S$ is transitive.

By 3.1, 3.2 and 3.3 $S$ is an equivalence relation.