# Proofs study problems

### Problem 1

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.

Hints

Model solution

### Problem 2

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).

Hints

Model solution

### Problem 3

For any real numbers x and y, let's define
the operation \(\oslash\) by the equation
\(x \oslash y = 2(x+y)\)

Disprove the following claim:
Claim: For any real numbers x,y, and z,
\((x \oslash y) \oslash z = x \oslash (y \oslash z)\)

Hints

Model solution

### Problem 4

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 \(F(p,t) \ge F(p,r)\).

Hints

Model solution