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missavvy
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Homework Statement
On the set of integers, define the relation R by: aRb if ab>=0.
Is R an equivalence relation?
Homework Equations
The Attempt at a Solution
R is an equivalence relation if it satisfies:
1) R is reflexive
Show that for all a∈Z, aRa.
Let a∈Z. Then if a is a negative integer, aa>=0. If a is a positive integer, aa>=0. And if a = 0, aa>=0.
Hence aRa
I feel like it is too simple.. lacking something??
2) R is symmetric
Show that for all a∈Z, aRb --> bRa
Let a∈Z, b∈Z such that aRb. By the definition of R, ab>=0.
This is not symmetric. Take a = -1, b = 2.
Then we have ab = -2 which is not >= 0.
3) R is transitive
if aRb and bRc implies aRc for all a,b,c ∈ Z
Let a, b, c ∈ Z s/t aRb, bRc --> aRc
Now I think this one is true.. but I'm not sure. But since aRb, and bRc, then you would always have ab or bc >=0 yea? so that means aRc must be true..
How would I prove it properly if it is correct?
Any help is appreciated! :) Thanks.