Please I need answer to these questions (8,9,10 and 11)

Thank you

8. The domain of discourse is the members of a chess club. The predicate B(x, y) means that person x has beaten person y at some point in time. Give a logical expression equivalent to the following English statements. (10 pts) a. No one has ever beat Nancy. b. Everyone has been beaten before. c. Everyone has won at least one game. d. No one has beaten both Ingrid and Dominic. e. There are two members who have never been beaten. 9. Translate each of the following English statements into logical expressions. The domain of discourse is the set of all real numbers. (6 pts) a. The reciprocal of every positive number is positive b. There is no smallest number c. There are two numbers whose ratio is less than 8:41 1 instructure-uploads.s3.amazonaws.com 10. Write the negation of each of the following logical expressions so that all negations immediately precede predicates. (8 pts) b.3x3y P(x, y) ^ Vx Vy Q(x, y) c.3xYy ( P(x, y)- P(y, x) ) d.3xYy ( P(x, y)-Q(x, y) ) 11. Use a truth table to prove the conclusion from the hypotheses. The hypotheses are: (10 pts) a. If I drive on the freeway, I will see the fire. b. I wl either drive on the freeway or take surface streets c. I am not going to take surface streets. Conclude that I will see the fire. Use the following variable names: p: I drive on the freeway r: I take surface streets .q: I see the fire