Question #128805
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
Expert's answer
2020-08-10T18:06:39-0400

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

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

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

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

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

2. RR is total.

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

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

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

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

By 3.1, 3.2 and 3.3 SS is an equivalence relation.


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