Use theorem 2.1.1 to verify the logical equivalences in 50–54. supply a reason for each step.

∼(p ∨ ∼q) ∨ (∼p ∧ ∼q) ≡ ∼p

Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of 1.1. (round your answers to four decimal places.) (a) if the distribution is normal, what is the probability that the sample mean hardness for a random sample of 8 pins is at least 51?