Write down the inference rules (11) and (AE). Then give natural deduction proofs of the following: (a) (p ^ q) F (q 1 (p 1 p)) (b) p, (q 1 -r) F (r 1 p) Definition 2.7 (The deduction rules for ^). 014 0 A4 (1 E) (AE) o V (010) (11) (AE) If D is a natural deduction tree with root (6 Ay) where o and are any propositional formulas then D -(1 E) and D - (1 E) are natural deduction trees. (A1) If D. is a natural deduction tree with root o and D2 is a natural deduction tree with root y then D D2 - (1 I) is a natural deduction tree. (6 A) With just these rules we cannot deduce much, but from the premise o Ay we can deduce y 10. 0 AV 0 AU (AE) (AE) o (NI) U 10 Write down the inference rules (11) and (AE). Then give natural deduction proofs of the following: (a) (p ^ q) F (q 1 (p 1 p)) (b) p, (q 1 -r) F (r 1 p) Definition 2.7 (The deduction rules for ^). 014 0 A4 (1 E) (AE) o V (010) (11) (AE) If D is a natural deduction tree with root (6 Ay) where o and are any propositional formulas then D -(1 E) and D - (1 E) are natural deduction trees. (A1) If D. is a natural deduction tree with root o and D2 is a natural deduction tree with root y then D D2 - (1 I) is a natural deduction tree. (6 A) With just these rules we cannot deduce much, but from the premise o Ay we can deduce y 10. 0 AV 0 AU (AE) (AE) o (NI) U 10