Discrete Mathematics Answers

Solve the recurrence relation an=3an-1-3an-2-an-3 n>=3 initial conditions are a0=1 a1=-2 a2=-1

Write using predicates and quantifiers.

(1)   For every m, n ∈ N there exists p ∈ N such that m < p and p < n.

(2)   For all nonnegative real numbers a, b, and c, if a 2 + b 2 = c 2 , then a + b ≥ c. 3.

(3)   There does not exist a positive real number a such that a + 1 a < 2.

(4)   Every student in this class likes mathematics.

(5)   No student in this class likes mathematics.

(6)   All students in this class that are CS majors are going to take a 4000 level math course. 

Suppose the universe of discourse for x is the set of all FSU students, the universe of discourse for y is the set of courses offered at FSU, A(y) is the predicate “y is an advanced course,” F(x) is “x is a freshman,” T(x, y) is “x is taking y,” and P(x, y) is “x passed y.” Use quantifiers to express the statements

(1)   No student is taking every advanced course.

(2)   Every freshman passed calculus.

(3)   Some advanced course(s) is(are) being taken by no students.

(4)   Some freshmen are only taking advanced courses.

(5)   No freshman has taken and passed linear algebra.

Suppose S(x, y) is the predicate “x saw y,” L(x, y) is the predicate “x liked y,” and C(y) is the predicate “y is a comedy.” The universe of discourse of x is the set of people and the universe of discourse for y is the set of movies. Write the following in proper English. Do not use variables in your answers.

(1)   ∀ y ¬S (Margaret, y)

(2)   ∃ y ∀ x L(x, y)

(3)   ∃ x ∀ y [C(y) → S(x, y)]

(4)    Give the negation for part 3 in symbolic form with the negation symbol to the right of all quantifiers.

(5)   state the negation of part 3 in English without using the phrase” it is not the case.”