Question: In each case either prove the result by splitting into cases, or give a counterexample. (a) If n is any integer, then n2 = 4k

In each case either prove the result by splitting into cases, or give a counterexample.
(a) If n is any integer, then n2 = 4k + 1 for some integer k.
(b) If n is any odd integer, then n2 = 4k + 1 for some integer k.
(c) If n is any integer, n3 - n = 3k for some integer k.

Step by Step Solution

★★★★★

3.46 Rating (162 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

b The implication here is p q where p is the statement n is any odd integer and q is the ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

950-M-L-A-L-S (6742).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

In each case either prove the assertion or give an example showing that it is false. (a) If A and B are both invertible, then A + B is invertible. (b) If A4 = 3I, then A is invertible. (c) If AB = B...

In each case either prove the statement or give an example showing that it is false: (a) If det A = 0, then A has two equal rows. (b) If R is the reduced row-echelon form of A, then det A = det R....

In each case either prove the statement or give an example showing that it is false: (a) If A is invertible and adj A = A-1, then det A = 1. (b) If det A 0 and AB = AC, then B = C. (c) If adj A = 0,...

1. A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, then one-half of the value that appears on the die. Determine her...

\fsh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m Math 280 Strategies of Proof Fall 2007 Class Notes 3.6 Proving Conditional Statements II: Indirect Proofs In Section 3.4, we studied...

\f3. Use any method to prove the statement. 1) For eyery integer n, if n2 is odd. then n is odd. 3. Write what you would suppose and what you would need to show to prove this statement by...

1. Recursion: Define a Recursive Integer List (RIL) to be a list of a finite number of objects, where each object is either an integer or is an RIL. For example, L = (3, 2, [4, 3], [[3,9), (), 8], 7]...

(i) Write down the linear program relaxation for the vertex cover problem and solve the linear program. [6 marks] (ii) Based on the solution of the linear program in (b)(i), derive an integer...

Comp 2700 (Discrete Structures) Fall 2016. Problem Set 1. Submissions: This assignment is due at 11:59 PM on the 7th of September, 2016. Each student must submit his or her own assignment. Solutions...

A seismic probe bores itself into the seabed, going as deep as it can before running out of fuel. This takes about five minutes. It rotates its spiral drill head at rate R(t) that follows a...

We purchase a truck for $10,000 We pay $2,000 down and put the rest on Accounts Payable How does the journal entry look

Garcia Corporation has the following accounts at December 31: Common Stock, $10 par, 5,000 shares issued, $50,000; Paid-in Capital in Excess of Par-Common Stock $32,000; Retained Earnings $45,000;...

The starting point in caiculating net operating income is the total annual income the property would produce assuming 1 0 0 % occupancy and no collection losses. This is commonly referred to as...

Using the data in MACRO.XLS and statistical software, which of the following variables appear to be nonstationary because of nonconstant means or nonconstant variances? Produce the graph for each...

Consider any right triangle with base b and height h, situated as shown. Show that the midpoint of the hypotenuse P is equidistant from the three vertices of the triangle. Determine whether each of...

Suppose that for some function h, h(x + 2) = 1 / 2 x. Find h(-2)

Find the center and the radius of the circle (x + 4)2 + (y - 5)2 = 36

Project 3: Stockholders' Equity (10 points) Required: Answer the questions on the following page in this booklet and turn in the entire booklet at the beginning of class on the date specified by your...

Which of the following pieces of code, ( a ) or ( b ) , would be more efficient ( run faster ) ? Justify your answer and state the name of one function you would use to verify the speed difference...

Exercise GOOD FAIR POOR SIZE OF MARKET MARKET MARKET FIRST STATION ($) (S) ($) Small 50.000 20,000 -10,000 Medium 80,000 30,000 -20,000 Large 100,000 30.000 -40,000 Very large 300,000 25,000 -160,000...