there is a student in this class who has taken every mathematics course offered at this school

What is discrete math?

A discrete mathematics class contains 1 mathematics major who is a freshman, 12 mathematics majors who are sophomores, 15 computer science majors who are sophomores, 2 mathematics majors who are juniors, 2 computer science majors who are juniors, and 1 computer science major who is a senior. Express each of these statements in terms of quantifiers and then determine its truth#N#value.#N#a) There is a student in the class who is a junior.#N#b) Every student in the class is a computer science major.#N#c) There is a student in the class who is neither a mathematics major nor a junior.#N#d) Every student in the class is either a sophomore or a computer science major.#N#e) There is a major such that there is a student in the class in every year of study with that major.

What is a statement in prenex normal form?

A statement is in prenex normal form (PNF) if and only if it is of the form#N#Q1x1Q2x2⋯QkxkP(x1, x2, … , xk)#N#where each Qi, i = 1, 2 , …, k, is either the existential quantifier or the universal quantifier, and P(x1, …, xk) is a predicate involving no quantifiers. For example, ∃x∀y(P(x, y) ∧ Q(y)) is in prenex normal form, whereas ∃xP(x) ∨ ∀xQ(x) is not (because the quantifiers do not all occur first).#N#Every statement formed from propositional variables, predicates, T, and F using logical connectives and quantifiers is equivalent to a statement in prenex normal form. Exercise 51 asks for a proof of this fact.#N#Show how to transform an arbitrary statement to a statement in prenex normal form that is equivalent to the given statement. (Note: A formal solution of this exercise requires use of structural induction, covered in Section 5.3.)

What is the function of quantifiers and connectives?

Use quantifiers and logical connectives to express the fact that every linear polynomial (that is, polynomial of degree 1 ) with real coefficients and where the coefficient of x is nonzero, has exactly one real root.

Is Qi a predicate?

where each Qi, i = 1, 2, …, k, is either the existential quantifier or the universal quantifier, and P(x1, …, xk) is a predicate involving no quantifiers. For example, ∃x∀y(P(x, y) ∧ Q(y)) is in prenex normal form, whereas ∃xP(x) ∨ ∀xQ(x) is not (because the quantifiers do not all occur first).