Background: To show two logical expressions are logically equivalent, you may.construct truth tables or-use the known list of logical equivalences on page.52 of the textbook. . .Also, double negative is available: ~(~X) = X. . To show they are not logically equivalent, it suffices to show one assignment. of truth values where the two expressions evaluate to different truth.values. 1 Example.1. We claim that.R A S is logically equivalent to ~ ((~R) V ( ~ S ) ) . 1 Using a truth table, the truth values in the columns for these expressions. are the same. 1 Sa RAS ~Rx ~ Sa ( ~ R ) V ( ~S ) A ~ (( ~ R ) V ( ~ S ) Ta Ta Fa Fa Ta Ta Fa Ta Fa F Ta Fa Ta Fa Fa Fa Ta Ta Ta Fa Alternatively, we can use DeMorgan with P = ~R, Q = ~S:1 ~ ( ( ~ R ) V ( ~ S ) ) - ( ~ ( ~ R ) ) 1 ( - ( - S ) ) - RAST The second equality is by double negatives. 1 Example.2. .We.claim that.P V (Q A R) and.(P v Q) A R are not logically. equivalent. 1 If we set.P = T, Q = T, R = F, then we have TV (T AF) = TVF =TH (T VT) AF = TAF = Fq and so the expressions evaluate to distinct truth values. They are not. logically equivalent. 17