Recent content by Ceci020

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    Undergrad Interpretation on the meaning of some graph theory statements

    Hello everyone, I'm studying basic graph theory, and my instructor gives me these statements to translate into pictures. I don't quite understand the meanings of the statements, but I have some thoughts about them. 1/ "Any two vertices a, b are connected by at least 2 distinct paths of...
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    Graduate Proving N is Maximal Ideal of G in Real Functions

    Thanks for your responses. Yes, I'm familiar with this idea. I think the goal of this problem is to show this, since ℝ is a field. ** By how the person, who put up this question, restates the question for me, I have some updates : 1/ Ring G indeed contains all real functions. There...
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    Graduate Proving N is Maximal Ideal of G in Real Functions

    Given: Denote a ring G = { all functions from ℝ --> ℝ } And set N = { all functions from ℝ --> ℝ such that for any x in ℝ, f(x) = 0 } Want to prove: To prove that N is the maximal ideal of G by showing that the quotient ring G / N is isomorphic to the set of real numbers ℝ ** What I'm...
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    Solving Prove: na < b & (n+1)a > b with No a Dividing b

    ** Homework Statement Question is to prove: If 1 < a < b and if a does not divide b then there exists n in N such that na < b and (n+1)a > b ** My thoughts: Since a does not divide b for 1 < a < b, I want to refer to the Euclidean algorithm, so: b = q1 * a + r1 for r1 is the first...
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    Proof that LCM(c,d) divides LCM(a,b), known GCD(a,b) divides GCD(c,d)

    ** Homework Statement 1/ set S = set of the multiples of any two natural numbers a, b S = {n in N such that a|n and b|n} 2/ Denote min(S) = LCM(a,b) = least common multiple of a and b From previous result, I already proved that : If a divides c and if b divides d, then GCD(a,b)...
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    Prove a set X is union of disjoint equivalence classes

    Homework Statement Prove: If E1, · · · , Ek are the disjoint equivalence classes determined by an equivalence relation R over a set X, then (a) X = union of disjoint equivalence classes Ej (b) R = union of disjoint (Ej x Ej) Homework Equations R is a subset of X x X The Attempt at a Solution...
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    Prove Relationship between Equivalence Relations and Equivalence Classes

    I'm not sure if I did these 2 questions correctly, so would someone please check my work for any missing ideas or errors? Question 1: Homework Statement Prove: For every x belongs to X, TR∩S(x) = TR(x) ∩ TS(x) Homework Equations The Attempt at a Solution TR(x) = {x belongs to X such that...
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    Equivalence Relation, prove dom(R) = range(R) = X

    I'm sorry but can you explain a little bit more? I think that since for all x in X, <x,x> is always in R holds, then according to the conditions of the domain, there indeed exists x E X such that there also exists a Y (namely, y = x) that makes <x,y> E R. But how about the range? thank you
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    Equivalence Relation, prove dom(R) = range(R) = X

    Homework Statement Given: R is an equivalence relation over a nonempty set X Prove: dom(R) = X and range(R) = XHomework EquationsThe Attempt at a Solution I have the following thoughts: About the given: Since R is an equivalence relation over X by hypothesis, R satisfies: Reflexivity: <x,x>...
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    Prove an Equivalence Relation R over N x N

    Homework Statement Given: A relation R over N x N ((x,y), (u,v)) belongs to R. i.e (x, y) ~ (u,v) If max(x,y) = max(u,v), given that max(x,y) = x if x >= y = y if x < y Prove that R is an equivalence relationHomework Equations I know that to prove an equivalence...