Discrete Maths :- Relations

Question

Check if the following relation is Antisymmetric,where R is defined on set of integers

R ={ (x,y) | y=$x^i$, for some i $\varepsilon$ Z}

Answer 1

Yes, the relation is anti-symmetric.

As xRy holds y=x^i should hold for every INTEGER.

now, for yRx to hold x=y^i also, which is not possible unless we invert the power, 1/i, the only case where 1/i will be an integer, is when i=1.

Which makes x=ywe can also check for typical cases, like (-1,1) which holds as 1=-1^2 but not vice versa for any i.

(x,0) and (0,y) where x and y are not equal to 0 itself cannot exist as only one value for x and y i.e. 0, which makes the only possible value as (0,0) which is reflexive and does not violate anti symmetry.

Therefore, xRy where y=x^i for some INTEGER i, is anti symmetric.

 

Not to mention that the case would have been different for Real Number domain of i.

Comments

 eyeamgj

isn't this answer wrong ???

when i =0, x=5, y=1

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(5,1) is in R due to 1 = 5⁰

is (1,5) also in R ?

that means, 5 = 1⁰ ===> is it true ? No, then (1,5) is not in R.

∴ R is anti-symmetric

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yeah bro.... i m sleeping in the morning. Woke up too early today

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@nephron

i hope, i slept when you woke up  :)