1. Construct a truth table for the following statement form, then state whether or not the given statement form is a tautology.
[p→(q→r)]↔[(~p)∨((~p)∨(~q)∨r)].
2. Write the negation of
(∀x ∈ Z)(∃y ∈ Z) such that x + y > 0.
Is the original statement true or false? Justify your answer.
3. Determine whether the following argument is valid.
If today is Friday, then I do not study.
If I do not study, then I party.
I party and today is not Friday.
Therefore I do study.
4. State whether or not the following statements are true or false. If the statement is true prove it, if it is false give a counter example.
(i) ∀a,b ∈ R √(a+b) = √a + √b.
(ii) ∀a,b ∈ Z, if 2a + b is even then a and b are even.
(iii) ∃a ∈ Z, s.t. (2a + 1)/2 is odd.
(iv) ∀n ∈ z, if n2 is even then n is even.