Help understanding equivalence relation

[smithnya]

Ok, I am barely beginning to understand the subject. I understand that the relation ≥ on N(naturals) is reflexive, not symmetric, and transitive. I don't understand why it is transitive though. Can someone explain?

Also, I understand why x² = y² is reflexive and symmetric, but I don't understand why it is transitive. Please help me out.

 

[digfarenough]

Transitivity means R(a,b) and R(b,c) implies R(a,c).

If R(a,b) means that a≥b and R(b,c) means b≥c then is it clear that a≥b≥c so a≥c and so R(a,c)?

If R(a,b) then a^2 = b^2. If R(b,c) then b^2 = c^2. Substitution implies that a^2 = c^2 so R(a,c). Hence transitivity.

Does that help or have I simply restated your question?

 

[smithnya]

Transitivity means R(a,b) and R(b,c) implies R(a,c).

If R(a,b) means that a>=b and R(b,c) means b>=c then is clear that a>=b>=c so a>=c and so R(a,c)?

If R(a,b) then a^2 = b^2. If R(b,c) then b^2 = c^2. Substitution implies that a^2 = c^2 so R(a,c). Hence transitivity.

Does that help or have I simply restated your question?

Ok, is it allowed to substitute values for x and y , or a and b, etc to facilitate?

 

[Bacle2]

Ok, is it allowed to substitute values for x and y , or a and b, etc to facilitate?

Yes; actually, the relation and its properties should hold for all substituted values from

the set you're working with.

 

[smithnya]

Yes; actually, the relation and its properties should hold for all substituted values from

the set you're working with.

So maybe that is where I am doing something wrong. For example:

1≥ 1 would make it reflexive

1≥2 and 2≥1 would make it not symmetric

but

1≥2
2≥3
and 1≥3 is clearly not true, so why is it transitive is the statements are false?

 

[digfarenough]

Your example
"1≥2
2≥3
and 1≥3"
is true if you change what ≥ means so that it means less than!

Say R(a,b) means that a≥b.
Then R(2,1) is true because 2≥1.
R(3,2) is true because 3≥2.

That means R(3,1) is true for two reasons:
A. R(3,1) is true from transitivity because R(2,1) and R(3,2)
B. R(3,1) is true by the definition because 3≥1

 

[smithnya]

Your example
"1≥2
2≥3
and 1≥3"
is true if you change what ≥ means so that it means less than!

Say R(a,b) means that a≥b.
Then R(2,1) is true because 2≥1.
R(3,2) is true because 3≥2.

That means R(3,1) is true for two reasons:
A. R(3,1) is true from transitivity because R(2,1) and R(3,2)
B. R(3,1) is true by the definition because 3≥1

But how can I just switch from ≥ to ≤? Isn't the crux of the problem that the relation be specifically ≥ on N?

 

[digfarenough]

I don't understand your question.

Transitivity means that if R(a,b) is true and also if R(b,c) is true then R(a,c) is true. If R(a,b) is not true or if R(b,c) is not true, then it says nothing about R(a,c).

Your example stated "1≥2" implying that you mean that statement is true and that "2≥3" is true. Neither of those are true if you mean ≥ means greater-than-or-equal, so they do not imply that "1≥3".

(Half joking, I said "if you change what ≥ means so that it means less than!" because you never did say what you mean by the symbol ≥, but since 1 is less than 2, your example would actually be true if you mean that ≥ means less than.)