Discrete Mathematics

Question

Let the logical operator ↑ be defined by the equivalence p↑q ≡¬(p ∨q).

(a) Find a compound proposition logically equivalent to ¬p using only the

logical operator ↑

(b) Find a compound proposition logically equivalent to p ^ q using only the

logical operator ↑

(c) Find a compound proposition logically equivalent to p ⇒ q using only the

logical operator ↑

Answer 1

(a) p↑p = ¬(p∨p) = ¬p

(b) (p↑p)↑(q↑q) = ¬p ↑ ¬q = ¬(¬p ∨ ¬q) = p ^ q

(c) ((p↑p)↑q)↑((p↑p)↑q) = ¬((p↑p)↑q) = ¬( ¬p↑q) = ¬(¬(¬p ∨ q)) = ¬p ∨ q = p ⇒ q