Given relations on S = {1, 2, 3, 4, 5}?

Question

Given relations on S = {1, 2, 3, 4, 5}

R1 is the <= (less than or equal) relation on S. In other words, (a, b) is in R1 if a <= b.
R2 is the < (less than) relation on S, and R3 is the = (equal to) relation on S.

Is any R1, R2, R3 relexive? symmetric? and transitive? Why?
Please any one help. I don't understand this question to explain it.

Answer 1

R2:
is transitive since (a,b) in R2 and (b,c) in R2, then a is not reflexive, since (a,a) is not in R2, because a=a and so a is not less than a.
is not symmetric since (a,b) is in R2, thus a
R1:
is transitive as above.
is reflexive, since a=a, so a leq a, thus (a,a) in R1
is not symmetric, since (1,2) in R1, but (2,1) is not in R1.


R3:
is transitive, as above.
is reflexive, as above.
is symmetric, since (a,b) in R3, thus a=b and b=a. Therefore (b,a) in R3.

Thus R3 is the only "equivalence relations," or a relation that is reflexive, symmetric, and transitive.

good question.

Answer 2

Okay, I'll see if I can help out.

Reflexive means that (1,1), (2,2) (3,3) (4,4) and (5,5) are all in the set. So, R1 and R3 should all be reflexive because 1 =1 2=2 etc and 1<= 1, 2<=2 ect. However, R2 would not be reflexive because 1 is not less than itself. (you only have to show one counter example).

Symmetric means that if (a,b) is in the set then (b, a) must also be in the set. So R1 And R2 are NOT Symmetric because Just because 1<2 or 1 <= 2 doesn't mean thatn 2<1 or 2<=1,. So, in R1 and R2 (1,2) are in the set but (2,1) is not. So they are not symmetric. Now in R3 it gets a little harder to explain. R3 is symmetric because if for instance if (a,b) is in the set then (b,a) must also be in the set.....but since R3 is the relation =, the only elements in this set are (1,1) (2,2) (3,3, (4,4) and (5,5)...So this means if (1,1) is in the set for it to by symmetric then (1,1) must also be in the set....understand.

Transitive means that if (a,b) and (b,c) are elements in the set then (a,c) must be in the set....So, R1, R2 and R3 (R3 for the same reason it is symmetric). Here is why R1 and R2 are symmetric.... if 1<2 and 2<3 then 1<3 True. This is the same reasoning for R2. So, if (1,2) and (2,3) are in R1 and R2 then (1,3) are also in the set.

I hope this is helpful. It is to explain if you are not in person.

So, R1, R3 are reflexive
R3 is symmetric
R1, R2, R3 are transitive.

Since R3 is reflexive, symmetric and transitive it is what is called an equivalence relation.

I hope this helps.

Answer 3

R is called reflexive if for every (a,b) in R then (b,a) in R
R is called symmetric if for every a in s then (a,a) in R
R is called transitive if (a,b) & (b,c) in r ==> (a,c) in R
So R1 is symmetric, transitive but not reflexive
R2 is only transitive
R3 is symmetric, transitive and reflexive

Answer 4

I am not sure that I understand the question at hand either but best of luck to ya!!!