1. Let P(x) be given by "x is a prime number," where the domain of x is the set of all positiveintegers less than 30. Recall that a prime number is an integer that is only divisible by 1 anditself. The integer 1 is not defined to be prime.(a) Is P(x) a propositional function? Explain.(b) Find all x such that P(x) is false.2. Let Q(x,y) be given by “x² + y² ≤ 4" where x, y € Z. Find all ordered pairs (x, y) forwhich Q(x, y) is true.

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Answered: Let P(x) be given by "x is a prime…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Determine if these functions are reflexive, symmetric, antisymmetric, and/or transitive. a. {(x, y) | x = ëyû} on the set of real numbersb. {(m, n) | m2 = n2 } on the set of integersc. {(a, b) | classroom a has a larger capacity than classroom b} on the set of classroomsd. {(1, 1), (2, 2), (1, 2), (2, 3), (3, 2), (3, 3)} on the set {1, 2, 3}
Let P(x, y) be "x has taken class y," and Q(x, y) be "x has passed class y" and M(y) be "y is a math class," where the domain of x consists of all students in your class and the domain of y consists of all classes at LU. Use quantifiers to express this statement. (Assume that all email messages that were sent are received.) Every student in your class has passed a class at Liberty University that is not a math class.
Consider the given predicate functions. P(n): "n is a prime number" E(n): "n is an even number" P and E both have domain = {nZ:n>1} Identify the TRUE statement(s). Explain your answer. I. En(E(n)^P(n)) II. Vn(E(n)VP(n)) III. \n(→E(n)→ P(n)) IV. En(-P(n)→→E(n)) O II only O I only O I and II only O II and III only O III only O I and IV only SO IV only O none
Given any set of 15 integers, must there be two that have the sameremainder when divided by 12? Prove using the pigeonhole principle. Eitherdescribe the pigeons, the pigeonholes, and how the pigeons get to the pigeonholes, or describe a function by giving its domain, co-domain, and how elements of the domain are related to elements of the co-domain.
Suppose f: R → R is a function from the set of real numbers to the same set with f(x) = x + 1. We write £ to represent fo f and fatl= fof. Is it true that £22o f = fo f?? Why? Is the set {g: R R | gof=fog} infinite? Why?
2. In each item below, you are given sets A and B, and a formula for a (non?)function f: A → B. In each case, determine whether or not the formula gives a well-defined function. Fully justify your answers. (a) A = Q = B, f(r) = a root in B of the polynomial X² - r (b) A = C = B, f(2)= a root in B of the polynomial X² – z (c) A = P(R) - {0}¹, B = R₁ ƒ(E) = the smallest element of E (d) A = P(R), B = R, ƒ(E) = the smallest element of E if E admits a smallest ele- ment, and f(E)= 0 otherwise A function f : A → B is injective (or one-to-one) if it satisfies the following condition: for all a₁, a2 € A, if f(a₁) = f(a₂), then a₁ = a₂; f is surjective (or onto) if f(A) = B, or, more explicitly, if for each b E B there exists a € A such that f(a) = b; f is bijective if f is both injective and surjective. Here are some examples. • The squaring function f : R → R given by f(x) = x² is not injective because f(-1) = 1 = f(1); f also fails to be surjective because -1 is not in the range of f. The…

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