Let Q denote the relation on the set Z of integers, where integers x
and y satisfy xQy if and only if x^{2} + 5y^{2} is divisible by 6.
Determine whether or not the relation Q is symmetric and whether or not
the relation Q is transitive.
Thank you Plato and Nehushtan!
I think I now understand?
If we suppose xQy then x^{2} + 5y^{2} is divisible by 6.
Then (x^{2} +5y^{2}) + (y^{2} + 5x^{2}) = 6(x^{2} + y^{2})
and then (y^{2} + 5x^{2}) = 6(x^{2} + y^{2}) - (x^{2} +5y^{2})
6(x^{2} + y^{2}) - (x^{2} +5y^{2}) is divisible by 6 since the difference of 2 integers divisible by 6 is itself an integer divisible by 6.
Thus yQx, showing that Q is symmetric.
If we suppose xQy and yQz then (x^{2} +5y^{2}) and (y^{2} + 5z^{2}) are both integers divisible by 6.
Then (x^{2} + 5z^{2}) = (x^{2} +5y^{2}) + (y^{2} + 5z^{2}) - 6y^{2}
It follows that (x^{2} + 5z^{2}) is divisible by 6.
Thus xQz, showing that Q is transitive.