(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Decide if the following are Equivalence relations and if so describe their classes

i) a[itex]\equiv[/itex] b if 2 divides a^2+b^2

ii) a[itex]\equiv[/itex] b if 2b[itex]\geq[/itex] a

2. Relevant equations

3. The attempt at a solution

ii) isnt an equivalence relation. it is reflexive but not symmetric. 2a [itex]\geq[/itex] b

i) Its reflexive as [itex]a^2[/itex]+[itex]a^2[/itex] = 2[itex]a^2[/itex] which is divisable by 2.

its symmetric

[itex]a^2[/itex]+[itex]b^2[/itex] = 2x

[itex]b^2[/itex]+[itex]a^2[/itex] = [itex]2a^2[/itex]+[itex]2b^2[/itex]-[itex]a^2[/itex]-[itex]b^2[/itex]

[itex]b^2[/itex]+[itex]a^2[/itex] = [itex]2a^2[/itex]+[itex]2b^2[/itex]-2x

[itex]b^2[/itex]+[itex]a^2[/itex] = 2[itex]a^2[/itex]+[itex]b^2[/itex]-x)

so its divisable by 2

its symetric

[itex]a^2[/itex]+[itex]b^2[/itex] = 2x

[itex]b^2[/itex]+[itex]c^2[/itex] = 2y

we need to show [itex]a^2[/itex]+[itex]c^2[/itex] = 2w

[itex]b^2[/itex] = 2y-[itex]c^2[/itex]

this gives[itex]a^2[/itex]+2y-[itex]c^2[/itex] = 2x

[itex]a^2[/itex]+[itex]c^2[/itex] = 2x-2y-[itex]2c^2[/itex]

[itex]a^2[/itex]+[itex]c^2[/itex] = 2(x-y-[itex]c^2[/itex])

so it is an equivalence relation. As for the classes are there infinity/2 classes? plus how do i describe them? if a & b are both even or both odd they are divisable by 2 but if a is odd and b is even or vise versa then it is not...

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# Another equivalence relation

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