Answer to Question #128805 in Discrete Mathematics for Chamali

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)\\}\\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.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS