Let the domain be the set of all animals. Define the following predicates: A(x) : x can add. W(x) : x can waltz. • D(x) : x is a dog. • C(x) : x is a cat. Translate each of the following logical expressions into English. 1. Vx(D(x) → (A(x) v W(x)). 2. Ex(C(x) A ¬A(x) A W(x)) 3. ¬3x(D(x) A A(x)) 4. Vx((D(x) V C(x)) → W(x))

buy A. Determine whether the following statements are always true, sometimes true. or never true. the Easy Milk Tea w its should 1. Itx-25-0, then x-5. 2 Ifx-27-0, then x-3 3. If 5x -45-0, then x-9. Ifx 10x--25, then x--5. 5. Ifx-3x+ 3x-1-64, then x-5. The Zero Product Property states that: If a and b are expressions, and ab-0, then a-0. 7. If the product of two numbers is zero, then at least one of 8. According to the Zero Product Property, If a and b are expressions, and ab-0, then b-0. the numbers is zero. ey Ma 10. In solving an equation by factoring, one side of the equation must be zero. 9. If (x+1) (xr+ 2)-12, then x+1-12 or a+2-12.

Please do diagram for (v) and (vi) and explain (i)

7. Variable represents students, y represents courses. M(y) rep. y is a math course, F(x) rep. x is a freshman, B(x) rep. x is a full-time student, T(x, y) rep. x is taking y. Write the statement in good English without using variables in your answers. a) VrayT(x, y) b) 3rVyT(x, y) c) Vray[(B(x) ^ F(x)) → (M(y) ^ T(x, y))]

Let à = ( − 4, — 5, 2) and b = (-1,0, 4). Find à x b. a - >

Let M(x,y) be the predicate "person x has seen movie y," where the domain for x is the set of all people and the domain for y is the set of all movies. Express the following English sentence as a symbolic proposition: "There is a movie that has not been seen by any people." x-M(x,y) ⒸyVx-M(x, y) x-M(x,y) yx M(x,y) -

(4) 2. Let P(X) = “x lives in Wahiawa” and the u.d. for x is all students at Leeward Community College. a. Translate into symbols, “There is at least one Leeward Community College student who lives in Wahiawa.” b. Using symbols, express the negation of your expression for 2a.

10] Javier works each weekend at an animal shelter where he earns $8 an hour. He is saving his earnings to go on a ski trip that costs $190. Number of Hours Javier's Work Record 10 9 8 0965 1 10 - Saturday |- Sunday Let h be the number of hours Javier worked. Select all the true statements. 10 2 Weekends □ Since 8h <190, Javier did not earn enough money to make the trip.

4. Translate the following sentences from English to predicate logic. The domain that you are working over is X, the set of people. You may use the functions S(x), meaning that " x has been a student of 6.042," A(x) meaning that "x has gotten an 'A' in 6.042," T(x) meaning that "x is a TA of 6.042," and E (x, y) meaning that "x and y are the same person." (i) There are people who have taken 6.042 and have gotten A's in 6.042. (ii) All people who are 6.042 TA's and have taken 6.042 got A's in 6.042. (iii)There are no people who are 6.042 TA's who did not get A's in 6.042.

Let a = (3, -3, 8), and b = (-7, −1, 3), and ☎ = (8, 4, 11). Find ||2a - 3b||.