Use direct proof to show that for all real numbers x and y, where x is not zero, if x and \(\frac{y+1}{3}\) are rational, then \(\frac{1}{x} + y\) is rational.
Prove the following claim:
Claim: For any real numbers x, y, s, t, m, n, where \(s \not = 0\), if xs = yt and ms = nt, then xn = ym.
Your proof must not use fractions (though you can cancel non-zero factors that occur on both sides of an equation).
\(x \oslash y = 2(x+y)\)Disprove the following claim:
Claim: For any real numbers x,y, and z, \((x \oslash y) \oslash z = x \oslash (y \oslash z)\)
If p and r are the precision and recall of a test, then the F1 measure of the test is defined to be \(F(p,r) = \frac{2pr}{p+r}\). Prove that, for all positive reals p, r, and t, if \(t \ge r\) then \(F(p,t) \ge F(p,r)\).