CramX
RegisterLog in

Mathematical Proofs And Induction: Set Theory, Even/Odd Properties, And Sum Of Cubes

Ace your assignment with this in-depth Assignment Solution on mathematical proofs.

Wyatt Evans
Contributor
4.3
44
7 months ago
Preview (2 of 3 Pages)
100%
Purchase to unlock

Page 1

Mathematical Proofs And Induction: Set Theory, Even/Odd Properties, And Sum Of Cubes - Page 1 preview image

Loading page ...

Mathematical Proofs and Induction: Set Theory, Even/Odd Properties,and Sum of CubesJeremy BreitMarch 25, 2013MTH 231Homework #2Problem:Prove that, for all integers n, if n2+ 3 is even, then n is odd.Solution:1. Assuming n2+ 3 is even, we can say that 2 divides the equation. n2+ 32|(𝑛2+3)βŸΊπ‘›2+3=2π‘˜π‘“π‘œπ‘Ÿπ‘ π‘œπ‘šπ‘’π‘˜πœ–β„€Simply,𝑛2+3is some integer multiple of 2.2. Since 2(k-2) is divisible by 2, and 2 is not divisible by 1, we can say thefollowing:𝑛2=2π‘˜βˆ’3=2(π‘˜βˆ’2)+1⟹2π‘‘π‘œπ‘’π‘ π‘›π‘œπ‘‘π‘‘π‘–π‘£π‘–π‘‘π‘’2(π‘˜βˆ’2)+1,⟹2π‘‘π‘œπ‘’π‘ π‘›π‘œπ‘‘π‘‘π‘–π‘£π‘–π‘‘π‘’π‘›2Simply, since 2 does not divide𝑛2,𝑛2cannot be even. This means that𝑛2is odd.3. Next we need to prove that since𝑛2is odd,𝑛is therefor odd. To do this, wewill prove the contrapositive. In this case, we will assume𝑛is even.𝑛=2π‘˜π‘“π‘œπ‘Ÿπ‘ π‘œπ‘šπ‘’π‘–π‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘˜Therefor,𝑛2=(2π‘˜)2=2(2π‘˜2)4. Since 2 divides𝑛2,𝑛2must be even. Therefor, is n is not odd,𝑛2is not odd aswell.5. Contrapositively, since𝑛2is odd,𝒏must be odd as well.

Page 2

Preview Mode

This document has 3 pages. Sign in to access the full document!

Study Now!

XY-Copilot AI
Unlimited Access
Secure Payment
Instant Access
24/7 Support
Document Chat

Document Details

Subject
Mathematics

Related Documents

View all
Exam Guides

AP Calculus AB Scoring Guide Unit 6 Progress Check

This math problem asks to find 𝑓(3) using values from a twice-differentiable function’s table. It tests understanding of derivatives and function behavior from given data points.

Solved Assignments

Clinical Psychologist

Grand Canyon University MAT-144 assignment β€œClinical Psychologist” by Steven Hughes (Nov 3, 2024). Includes grading notes and an β€œIncome and Projection” section with behavioral health salary estimates and a monthly expense budget table.

Class Notes

Module 4 Matrices

Overview of matrices: rectangular arrays of numbers, defining rows/columns and entries. Covers dimensions (mΓ—n), locating entries, types (square, row, column matrices), and using matrices for computations (e.g., soccer equipment costs).

Solved Assignments

MATH 1324 Homework 5 Answers

MATH 1324 Finite Math: Homework on compound interest. Calculate future value of a trust fund with 2.5% annual interest over 35 years. Learn to identify and apply the correct interest formulas.

Solved Assignments

Ap progress 1 5 all 5

Unit 5 Progress Check: MCQ Part A – A set of calculus multiple-choice questions covering the Mean Value Theorem, intervals of increasing/decreasing, finding local extrema from derivatives, and applying continuity/differentiability on given intervals.

Past Exams

The Fundamental Theorem of Calculus and Definite I

Practice definite integrals and apply the Fundamental Theorem of Calculus in this Skill Builder for Topic 6.7. Solve problems, cross off answers, and reinforce key AP Calculus concepts effectively

Past Exams

AP Calculus AB Scoring Guide

Accurate solutions to AP Calculus Unit 6 Progress Check: MCQ Part B. Covers integrals, antiderivatives, u-substitution, and graph analysis. Great for studying, checking your work, or exam prep. Clear answers with correct choices highlighted

Solved Assignments

Investigating the Rising Cost of MTA Transit Fare

Explore a math-based analysis of rising MTA transit fares over time. Use algebra, scatter plots, and linear modeling in Excel to investigate trends and predict future prices. Great for students learning real-world math applications.

Solved Assignments

Module 11 Geometry

Learn how to calculate the volume and surface area of a cube using simple geometry formulas. Includes clear examples and step-by-step explanationsβ€”perfect for Prealgebra and early Geometry students.

Past Exams

AP Calculus AB Unit 5 Progress Check Scoring Guide

Access AP Calculus practice questions, step-by-step solutions, and exam prep resources. Enhance your understanding with graphs, derivatives, and optimization problems in an interactive format. Perfect for high school and college review.