can u please answer number 8

111. 1. Give formal proofs of the following: 1. ( AB) V (AAC) from the premises A and B v C. (10) ii. A from the premise (AB)VA (10) {(AAC) V (DAC), B} = CAB (10) 2. Write down De Morgan's laws for quantifiers. (8) 3. Prove the following: 1. Vx (S(x) P(x)) -3x (S(x) 4- P(x)) il. Vx (S(x) +-P(x)) -3x (S(x) A P(x)) ili. 3x (S(x) A P(x)) -x (S(x) -P(x)) iv. 3x (S(x) A-P(x)) -VX (S(x) P(x)) (12) 4. Let P(x) be the statement "x spends more than five hours every weekday in class," where the domain for x consists of all students. Express each of these quantifications in English. (1) 3xP(x) (11) VxP(x) 3xP(x) (iv) VxP(x) (12) 5. Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. (1) No one is perfect. Not everyone is perfect. (111) All your friends are perfect. At least one of your friends is perfect. (12) 6. Given the following names and predicates: Max Claire Folly Silly Disk(x) Blank(x, t) Student(x) t