Description

Need help with a 4 math problems. Please find the attached file. Thank you for your help

1 attachmentsSlide 1 of 1attachment_1attachment_1

Unformatted Attachment Preview

MAU22C00: ASSIGNMENT 1

DUE BY FRIDAY, OCTOBER 16 BEFORE MIDNIGHT

UPLOAD SOLUTION ON BLACKBOARD

Please write down clearly both your name and your student

ID number on everything you hand in. Please attach a cover

sheet with a declaration confirming that you know and understand College rules on plagiarism. Details can be found on

http://tcd-ie.libguides.com/plagiarism/declaration.

1) (10 points) Please carry out the following proof in propositional logic

following the proof format in tutorial 1: Hypotheses: P → (Q ↔ ¬R),

P ∨ ¬S, R → S, ¬Q → ¬R. Conclusion: ¬R.

For each line of the proof, mention which tautology you used giving its

number according to the list of tautologies posted in folder Course Documents. Solutions based on truth tables or any other method except

for the one specified will be given NO CREDIT.

2) (10 points) Prove the following statement: If n is any integer, then

n2 − 3n must be even. (Hint: Cases come in handy here. See tautology

#26 for the basis of proofs by√cases. This proof follows the format of

the one given in lecture that 2 is not a rational number.)

3) (10 points) Prove via inclusion in both directions that for any three

sets A, B, and C

A ∩ (B C) = (A ∩ B) (A ∩ C).

Venn diagrams, truth tables, or diagrams for simplifying statements in

Boolean algebra such as Veitch diagrams are NOT acceptable and will

not be awarded any credit.

4) (10 points) Let N × N be the Cartesian product of the set of natural

numbers with itself consisting of all ordered pairs (x1 , x2 ) such that

x1 ∈ N and x2 ∈ N. We define a relation on its power set P(N × N) as

follows: ∀A, B ∈ P(N × N) A ∼ B iff (A B) ∪ (B A) = C and C

is a finite set. Determine whether or not ∼ is an equivalence relation

and justify your answer by checking each of the three properties in the

definition of an equivalence relation. Please note that a set C is finite

if it has finitely many elements. In particular, the empty set ∅ has zero

elements and is thus finite.

Purchase answer to see full

attachment

Tags:

Propositional logic

boolean algebra

tautology

equivalence relation

the Cartesian product

User generated content is uploaded by users for the purposes of learning and should be used following Studypool’s honor code & terms of service.

## Reviews, comments, and love from our customers and community:

This page is having a slideshow that uses Javascript. Your browser either doesn't support Javascript or you have it turned off. To see this page as it is meant to appear please use a Javascript enabled browser.