a) Show that the following logical equivalences hold for the Peirce arrow↓, where P ↓Q = ~ (P ∨ Q). P ∨ Q = (P ↓ Q) ↓ (P ↓ Q) P ∧ Q= (P ↓ P) ↓ (Q ↓ Q) b) Show that for the Shuffer stroke | P ∧ Q = (P | Q) | (P | Q) c) Use the result from (b) and example 2.4.7 from book to write P ∧ (∼Q ∨ R) using only Shuffer strokes

(a) By the definition of piece arrow-

  $P\downarrow Q=$ ~$(P\lor Q)$

  $P\downarrow Q$ =~$(P\lor P)$

  We have derived that $P\downarrow P$ is logically equivalent with ~P

      ~$P=P\downarrow P$

(b)$(P\downarrow Q)\downarrow (P\downarrow Q)$ =(~($P\lor Q))\downarrow$ (~$(P\lor Q)$

                       $=(P\lor Q)\land (P\lor Q)\\ =P\lor Q$

(c)$(P\downarrow P)\downarrow (Q\downarrow Q)$ =(~($P\lor P))\downarrow$ (~($Q\lor Q))$

                       $=(P\lor P)\land (Q\lor Q)\\ =P\land Q$