Hello guys :) I have this problem:
Let (x,y) in QxQ,
T: QxQ------------> IR
T(x, y)= Pi x+ y
Is this function surjective? I think it's not.... but i don' t know how to prove it :(
Let (x,y) in QxQ,
T: QxQ------------> IR
T(x, y)= Pi x+ y
Is this function surjective? I think it's not.... but i don' t know how to prove it :(
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T is not surjective.
For instance, there is no (x, y) in QxQ such that T(x, y) = πx + y = π^2.
(This follows from π being a transcendental number; that is π does not satisfy any polynomial equation with rational coefficients. More specifically, if x, y in Q did exist, then π is a zero of t^2 - xt - y, which contradicts the fact that π is transcendental.)
I hope this helps!
P.S.: I like the cardinality proof as well.
For instance, there is no (x, y) in QxQ such that T(x, y) = πx + y = π^2.
(This follows from π being a transcendental number; that is π does not satisfy any polynomial equation with rational coefficients. More specifically, if x, y in Q did exist, then π is a zero of t^2 - xt - y, which contradicts the fact that π is transcendental.)
I hope this helps!
P.S.: I like the cardinality proof as well.
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QxQ is countable, so T(QxQ) is also countable. IR is not countable, so T is not surjective.