For the following pairs of formulas, determine whether they are logically equivalent. justify your answer. i. (pv-9), -(p +9) ii. (p + ( p →q iii. ((p vq) ^ ( -p (qar) iv. (ph ( (r + ( p a)) v. ((p +r)^(q + ((p v q) +r)

Astudy was interested in determining if eating milk chocolate lowered someone's cholesterol levels.ten people's cholesterol was measured. then, each of these individuals were told to eat 100g of milk chocolate every day and to eat as they normally did. after two weeks, their cholesterol levels were measured again. is there evidence to support that their cholesterol levels went down? how should we write the alternative hypothesis? (mud = the population mean difference= before - after)a. ha: mud = 0b. ha: mud > 0c. ha: mud < 0d. ha: mud does not equal 0