# 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.

### 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).

### 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, $$(x \oslash y) \oslash z = x \oslash (y \oslash z)$$

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