The question I am trying to answer is:
Prove or disprove (that is prove it is false) each of the following statements. R is the set of real numbers.
• ∀x ∈ R, ∃y ∈ R, x + y > 0
• ∃x ∈ R, ∀y ∈ R, x + y > 0
• ∀x ∈ R, ∃y ∈ R, xy > 0
• ∃x ∈ R, ∀y ∈ R, xy > 0
• ∀x ∈ R, ∃y ∈ R, xy ≥ 0
• ∃x ∈ R, ∀y ∈ R, xy ≥ 0
I was able to do the first 4, which I got true, false, false, false, respectively. But I am having difficulties with the last two. I am also finding that when doing these I tend to assume that they are false, even if they are true. Can someone explain the last two as well as any tips when dealing with these types of problems?