I was wondering if anyone can look over my work, and tell me if i am doing everything correctly? link to the pdf file is below.

hw1(2).pdf - File Shared from Box.net - Free Online File Storage

Problems 1 and 2: I have not checked every cell in the truth table, but the conclusions are correct. In #2, write explicitly the answer to the question whether P ≡ Q.

Problem 3. You need a more precise proof that x > 1 -> x^2 > x. If x > 1, then x - 1 > 0 and x > 0, so x^2 - x = x(x - 1) > 0 as the product of two positive numbers.

Problem 4. Correct, though the formula is unusual. Usually, one has either $\exists x\,(x>1\land\dots)$ or $\forall x\,(x>1\to\dots)$. This is not to say that the original formula cannot occur somewhere.

Problem 5. $(y+1)^2 < y + 1$ can happen when -1 < y < 0. If you choose a different value of x, you'll need a more accurate proof as well. The overall answer is correct.

Problem 6.
to prove this statement is false, we need to show that for any arbitrary real number y , ∃x(x^2 < y + 1) is false
You need to find only one y. Your proof that x^2 < -2 is impossible is hard to read because it has to be written with formulas. However, in this case no further proof is necessary because $\forall x\; x^2\ge 0$ is a well-known fact.

It may be a good idea to review how to solve quadratic inequalities.