Let S(x, y) denote the statement "x has seen y" and D denote the set of all students in our class and M be the set of all movies. a. Express the following English sentence as a quantified proposition using the definitions above:
"For every movie there is a pair of students in our class who have both seen it." Hint: Use three quantifiers - one for the movie and two for a pair of students.

b. Negate the quantified proposition you wrote for part (a) (i. e., place a "¬" in front of it). Use de Morganâs law for quantifiers to move the negation inside the quantifiers.
c. Translate you answer for part (b) back to plain English.

Select all expressions that are equivalent to 2(3x + 7y). question 1 options: 6x + 14y 6x + 7y 1(6x + 14y)

Prove that 16^4-2^13-4^5 is divisible by 11.

(50 ) mrs. julien’s and mrs. castillejo’s classes are selling cookie dough for a school fundraiser. customers can buy packages of macadamia nut chip cookie dough and packages of triple chocolate cookie dough. mrs. julien’s class sold 25 packages of macadamia nut chip cookie dough and 30 packages of triple chocolate cookie dough for a total of $221.25. mrs. castillejo’s class sold 5 packages of macadamia nut chip cookie dough and 45 packages of triple chocolate cookie dough for a total of $191.25. (a) write the system of equations that model the problem. be sure to explain which equation represents which situation. (b) find the cost of each type of cookie. show your work. (c) explain which method you used to solve the system and why you chose that method.