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k3k3
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Homework Statement
Let x and y be irrational numbers such that x-y is also irrational.
Let A={x+r|r is in Q} and B={y+r|r is in Q}
Prove that the sets A and B have no elements in common.
Homework Equations
The Attempt at a Solution
Since x and y are in A and B, then x+r[itex]_{1}[/itex]-y-r[itex]_{2}[/itex] is irrational.
Using contradiction, assume their sum is z and z is an element of A[itex]\cap[/itex]B
x-y=z
Then x+r[itex]_{1}[/itex]-y-r[itex]_{2}[/itex]=z
Since z is in both A and B, then z can also be said to be x+r[itex]_{1}[/itex]+y+r[itex]_{2}[/itex]
Then x+r[itex]_{1}[/itex]-y-r[itex]_{2}[/itex]=x+r[itex]_{1}[/itex]+y+r[itex]_{2}[/itex]
Solving for the right side,
0=2(y+r[itex]_{2}[/itex])
But since the sum is irrational and 0 is not irrational, A and B cannot contain the same elements.
Am I thinking about this correctly?