Negation and Classical Proof Techniques - Science & Maths Assignment Help

Assignment Task

Rehearsal Set
1. Find a logically equivalent statement where negation only appears immediately in front of sentential variables (like p, q , r) and predicates (like P(x, y), A(t)).
(a) ¬(p ? ?k : ((¬q) =? (?x : R(x, k))))
(b) ¬((?x : P(x)) ? (?x : (P(x) =? Q(x))))

2. The following theorem statement contains only one implication. Rewrite the theorem statement (in plain English) in contrapositive form. It might help you to make an AST, but this is not strictly necessary.
For all integers a, b, and c, if a
2 + b
2 = c
2
then a, b, and c cannot all be odd.
You do not need to prove the theorem (the easiest proof involves modular arithmetic, which we do not yet know).

3. Set up a proof by contradiction for the following theorem. You do not need to prove the theorem. If p is a prime number and p / (ab) for some integers a and b, then either p / a or p / b.

4. The following theorem contains only one disjunction. Rewrite the theorem statement (in plain English) in implication form. It might help you to make an AST, but this is not strictly necessary.
If p is a prime number and p / (ab) for some integers a and b, then either p / a or p / b.
You do not need to prove the theorem.

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