Purchase Solution

Defining Certain Subsets of the Natural Numbers

Not what you're looking for?

Ask Custom Question

Let L be the language of addition (with equality) in first-order logic. That is, let L be the first-order language that allows for use of the equality symbol ("=") and whose only non-logical symbol is a binary function symbol "+". (That is, L has no constant symbols and no predicate (relation) symbols.)

Now consider the L-structure that has as universe the set N of all the natural numbers (0, 1, 2, ...) and where the function symbol "+" is interpreted as the usual addition on the natural numbers.

(a) Use an L-formula of first-order logic to define the set E_1 of all the even natural numbers.

(b) Use an L-formula of first-order logic to define the set E_2 of all the even natural numbers that are not divisible by 4.

(c) Use an L-formula of first-order logic to define the set S={1} (i.e., S is the one-element subset of the natural numbers whose only element is the number 1).

Purchase this Solution
Solution Summary

A certain language of first-order logic is given, and formulas in that language are then used to define certain subsets of the set of all the natural numbers. In addition to provision of the actual formulas, a detailed explanation of the choice of those formulas is provided.

Solution provided by:
Georgia Martin, PhD
Education
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
Recent Feedback
  • "Thanks for your assistance. "
  • "Thank you. I understand now."
  • "Super - Thank You"
  • "Very clear. I appreciate your help. Thank you."
  • "Great. thank you so much!"
Purchase this Solution
Free BrainMass Quizzes
Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Probability Quiz

Some questions on probability

View More Free Quizzes
Related BrainMass Solutions

  • Mathematics - Problems with Sets

    Using the formula, we can write the number of proper subsets for the given sets. 1. The set of natural numbers less than 10 The set of natural numbers less than 10 contains 9 elements namely 1, 2, 3, ... 9.

  • Mathematics - Combinatorial Mathematics

    Answer: Notation:- N = Set of natural numbers, W = Set of whole numbers, Z = Set of integers, Q = Set of rational numbers and R = Set of real numbers {1, 2, 3, 4, 5, 6, 7, 8, 9} is a subset of N, the set of natural numbers.

  • Sets and subsets

    P(N) is the set of all subsets of natural numbers.

  • Set of all finite and cofinite subsets of N

    422960 Set of all finite and cofinite subsets of N A subset A of the set N of all natural numbers is defined to be cofinite if N - A (the complement of A in N) is finite.

  • If A ( B and B ( C, can you conclude that A ( C ? can you conclude that A ( C?

    write the following in roster form: set j is the set of natural numbers between one and seven 6 state whether set A and B are equal, equivalent, both or neither.

  • Well-Ordering Axiom - Strong Induction

    First, note that if a subset of the natural numbers has only one element, a, then it certainly has a smallest element, namely a. Therefore, P(1) is true. Second, suppose that all subsets of size 1 through n each have a smallest element.

  • Recurrence Relations : Stirling Numbers

    156060 Recurrence Relations : Stirling Numbers A Stirling number Snk is definedc as the number of ways of partitioning the set of positive numbers (1,2,3,....n) into k non empty subsets.

  • Recurrence Relations : Stirling Numbers

    155856 Recurrence Relations : Stirling Numbers A stirling number Snk is defined as the number of ways of partitioning the set of positive numbers (1,2,3......n) into k non empty subsets.

  • Power set of a countably infinite set

    Let i be the natural number such that T corresponds to i (in the one-one correspondence between S and the set of all natural numbers). [Such i exists, because T is an element of S.]

View More Related Solutions