SKE Algo II (Winter 2019)/Homework 0/Problem set - Revision history

Srakrn: /* Revisiting proofs 2 */

2020-02-08T05:49:31Z

Revisiting proofs 2

Srakrn at 05:48, 8 February 2020

2020-02-08T05:48:46Z

Srakrn: Srakrn moved page SKE Algo II (Winter 2019)/Homework 0 to SKE Algo II (Winter 2019)/Homework 0/Problem set without leaving a redirect

2020-02-07T07:54:00Z

Srakrn moved page SKE Algo II (Winter 2019)/Homework 0 to SKE Algo II (Winter 2019)/Homework 0/Problem set without leaving a redirect

Srakrn at 16:49, 4 February 2020

2020-02-04T16:49:13Z

Srakrn: Created page with "= Introduction = == Grading policy == * '''Intuition based, not solution based:''' This homework will be graded based on your attempt and not your solution. Giving a complet..."

2020-02-04T16:38:39Z

Created page with "= Introduction = == Grading policy == * '''Intuition based, not solution based:''' This homework will be graded based on your attempt and not your solution. Giving a complet..."

New page

= Introduction =

== Grading policy ==

* '''Intuition based, not solution based:''' This homework will be graded based on your attempt and not your solution. Giving a completely wrong answer will '''not''', and never make your score zero.
* '''Think of them as an long-term brain teaser:''' Don’t complete them in one day, or you’ll missed the chances of perfecting your understandings and perspective of the problems little by little.

== Ethics and Code of Conduct ==

* '''Zero tolerance on plagiarism:''' This course enforces a zero tolerance policy on plagiarism.
* '''Wrong answer is better than correct but copied answer:''' The scoring criteria is designed to heavily penalise copied answers.
* '''No submission is better than turning in copied submission:''' The scoring criteria is designed to motivate students to submit their original work.
* '''Feel free to discuss:''' Discussion among friends and colleagues are completely normal, and we encourage that. However, please '''indicate your collaborators''', as it will be used to track works’ originality.

= Revisiting proofs 1 =

''These assignments were adapted from '''Discrete Mathematics''', Douglas E. Ensley and J. Winston Crawley, Wiley.''

== Implication A ==

Prove or disprove that for every integer <math>n \geq 1</math>, if <math>n</math> is odd then <math>n^2 + 4</math> is a prime number.

== Implication B ==

Prove or disprove that for every positive integer <math>n</math>, <math>2n^3</math> is divisible by 3.

= Revisiting proofs 2 =

''These assignments were adapted from '''Discrete Mathematics''', Douglas E. Ensley and J. Winston Crawley, Wiley.''

== Implication C ==

Prove or disprove that for every positive integer <math>n</math>, if <math>n</math> is odd then <math>n^3-n</math> is divisible by 4.

== Implication D ==

Prove or disprove that for all integers <math>n > 4</math>, if <math>n</math> can be expressed in the term of <math>z^2</math> for any integers <math>z</math>, then <math>n-1</math> is not a prime number.

== Implication E ==

Prove or disprove that for all integers <math>n</math>, if <math>n^2</math> is even then <math>n</math> is even. ''Hint: Use contrapositive.''

= Revisiting proofs 3 =

''Parts of these assignments were adapted from '''Discrete Mathematics''', Douglas E. Ensley and J. Winston Crawley, Wiley.''

== Exhaust A ==

Proof that if an integer <math>n</math> can be written in the form of <math>z^3</math> for any integers <math>z</math>, then it must satisfy one of the following critera:

* <math>n</math> must be a multiple of 9,
* <math>n</math> must be 1 more than a multiple of 9, or
* <math>n</math> must be 1 less than a multiple of 9.

== Exhaust B ==

Prove or disprove that for any integer <math>n</math>, <math>n^2 + 2 </math> is even.

== Exhaust C ==

Prove or disprove that every integer not divisible by 3 had a square that is of the form <math>3k + 1</math>.

= Revisiting proofs 4 =

In this section, use proof by induction to prove the given statement. Make sure to clearly '''indicate your steps of induction''' along with the '''induction hypothesis (I.H.)'''.

== Induction A ==

Prove that <math>n^2 + n</math> is even for any positive integers <math>n</math>.

== Induction B ==

Prove that <math>\sum_{i=1}^{N} i = \frac{(n)(n+1)}{2}</math> for any positive integers <math>N</math>.

== Induction C ==

Prove that <math>\sum_{i=1}^{N} i = \frac{(n)(n+1)(2n+1)}{6}</math> for any positive integers <math>N</math>.

← Older revision		Revision as of 05:48, 8 February 2020
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	== Induction C ==		== Induction C ==

−	Prove that <math>\sum_{i=1}^{N} i = \frac{(n)(n+1)(2n+1)}{6}</math> for any positive integers <math>N</math>.	+	Prove that <math>\sum_{i=1}^{N} i^2 = \frac{(n)(n+1)(2n+1)}{6}</math> for any positive integers <math>N</math>.

← Older revision		Revision as of 16:49, 4 February 2020
Line 1:		Line 1:
−	~~= Introduction =~~
−
−	~~== Grading policy ==~~
−
−	* '''Intuition based, not solution based:''' This homework will be graded based on your attempt and not your solution. Giving a completely wrong answer will '''not''', and never make your score zero.
−	* '''Think of them as an long-term brain teaser:''' Don’t complete them in one day, or you’ll missed the chances of perfecting your understandings and perspective of the problems little by little.
−
−	~~== Ethics and Code of Conduct ==~~
−
−	* '''Zero tolerance on plagiarism:''' This course enforces a zero tolerance policy on plagiarism.
−	* '''Wrong answer is better than correct but copied answer:''' The scoring criteria is designed to heavily penalise copied answers.
−	* '''No submission is better than turning in copied submission:''' The scoring criteria is designed to motivate students to submit their original work.
−	* '''Feel free to discuss:''' Discussion among friends and colleagues are completely normal, and we encourage that. However, please '''indicate your collaborators''', as it will be used to track works’ originality.
−
	= Revisiting proofs 1 =		= Revisiting proofs 1 =

@@ Line 23: / Line 23: @@
 Prove or disprove that for all integers <math>n > 4</math>, if <math>n</math> can be expressed in the term of <math>z^2</math> for any integers <math>z</math>, then <math>n-1</math> is not a prime number.
-== Implication E ==
+== Optional question: Implication E ==
 Prove or disprove that for all integers <math>n</math>, if <math>n^2</math> is even then <math>n</math> is even. ''Hint: Use contrapositive.''

← Older revision	Revision as of 07:54, 7 February 2020
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