1.1 Since {(0,0),(1,1),(2,2)}⊂R , then R is reflexive.
1.2 For each pair (a,b)∈R if (b,a)∈R then a=b , hence R is a symmetric relation.
1.3 If (a,b)∈R and (b,c)∈R , then (a,c)∈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 (≤) 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)∈S if and only if a=b , hence
3.1 Since {(0,0),(1,1),(2,2)}⊂S, then S is reflexive.
3.2 If (a,b)∈S , then a=b and (b,a)∈S , hence S is a symmetric relation.
3.3 If (a,b)∈S and (b,c)∈S, then a=b=c and (a,c)∈S , hence S is transitive.
By 3.1, 3.2 and 3.3 S is an equivalence relation.
Comments