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  1. K

    Prove Set of all onto mappings from A->A is closed

    Homework Statement Prove that set of all onto mappings of A->A is closed under composition of mappings: Homework Equations Definition of onto and closure on sets. The Attempt at a Solution Say, ##f## and ##g## are onto mappings from A to A. Now, say I have a set S(A) = {all onto mappings of A...
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    Show that is not onto (##\frac{x}{x^2+1}##)

    Homework Statement I need to show that $$\frac{x}{x^2+1}$$ is either onto or not. My domain is $$R-{0}$$ and range is $$R$$ Homework Equations I have learn to do this to show that a function is surjective y = $$\frac{x}{x^2+1}$$ and solve for x, but I am not sure how to proceed here. The...
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    Bohr-Sommerfeld Rule

    Homework Statement Imagine that force for is atom was ## F= - \frac{\beta}{r^4}##, rather than ##F=- \frac{ke^2}{r^2}##, and consider only circular orbits, it would remain true that ##L_n= n \hbar## a.) From Netwon's law find the relationship between ##T ##(Kinetic Energy) and ##V##, b.) Find...
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    Ambiguous GRE Question?

    Homework Statement Title of pie chart: New Construction in Daisy Hill Subdivision Given a pie chart with sections: Currently Completed: 26% Currently Under Construction: 42% Approved, but Not Yet Started: 32% Question: When construction is completed in Daisy Hill Subdivision there'll...
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    Classical Relativity and the Speed of Light

    Homework Statement Let's assume that the classical ideas of space and time are correct, so that there could only be one frame, "ether", in which light traveled with same speed in all directions. Assume that the earth's speed relative to the ether frame is our orbital speed around the sun...
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    Relativity of Orientation & Origin

    Homework Statement At time t=0, a block is released from point O on the slope shown in the figure. The block accelerates down the slope, overcoming sliding friction. a.) Choose axes 0xy as shown, and solve the equation ##\Sigma F = m a## into its x and y components. Hence find the block's...
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    Final Undergraduate Semester Guidance

    Hello all, I have a bit of a dilemma. I have two free electives to take and I am considering taking two "advanced" math classes and get a math minor or two semi-good classes from whatever. If I take two math classes and I get A's, it would be lovely. But, If I get low grades, my GPA will...
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    Epsilon- Delta Proof

    Homework Statement Prove that ## lim_{x\implies 1} \frac{2}{x-3} = -1 ## Use delta-epsilon. The Attempt at a Solution Proof strategy: ## | { \frac{ 2}{x-3} +1 } | < \epsilon ## ## \frac{x-1}{x-3} < \epsilon ## , since delta have to be a function of epsilon alone and not include x. I...
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    Reflexivity Implies Symmetry?

    Homework Statement Is this relation, R, on ## S= \{ 1, 2, 3 \} \\ R = \{ (1,1), (2,2) , (3,3) \}## Symmetric? It is obvious that it is reflexive.
  10. K

    Prove Divisibility

    Homework Statement a.) Prove: If an integer ##a## does not divide ##bc##, then ##a## does not divide ##b## and ##a## does not divide ##c##. b.) State and either prove or disprove the converse of the above statement. The Attempt at a Solution a.) Proof by contrapositive ## a|c \vee a|b...
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    Can I use induction?

    Homework Statement Prove for any ##a,b \in \mathbb{R^+} : |a-b| \leq \sqrt{a^2 +b^2} ## The Attempt at a Solution How should I start? Can I use induction? Should I use contrapositive?
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    Co-Primes Proof

    Homework Statement Let ##n \in \mathbb{Z} , n \not | 3##. Prove that ##gcd( n , n +3 ) =1 ## The Attempt at a Solution If n is not divisible by 3, then n = 3k+1 or n =3k+2 , ## k \in \mathbb{Z} ## What is a feasible approach? Can I do this? For first case, ## gcd(3k+1, 3k+4 ) = 1 \\...
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    Show F is Injective & Cardinality of Domain

    Homework Statement Let ## S = \{ (m,n) : m,n \in \mathbb{N} \} \\ ## a.) Show function ## f: S -> \mathbb{N} ## defined by ## f(m,n) = 2^m 3^n ## is injective b.) Use part a.) to show cardinality of S. The Attempt at a Solution a.) ## f(a,b) = f(c, d ) ; a,b,c,d \in \mathbb{N} \\\\ 2^a...
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    Prove Existence of Real Number

    Homework Statement a.) Prove ## \exists x, x \in \mathbb{R} | x^3 -x^2 = 5## I know that x = 2.1163, but how do I find this without a calculator? b.) Prove that ## \not \exists x, x \in \mathbb{R} | x^4 - 2x^2 +2 =0## The Attempt at a Solution x^2 ( x-1 )=5 for part a)...
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    Prove Same Cardinality (1,3) and [1,4]

    Homework Statement Prove that the open interval (1,3) and the closed interval [1,4] have the same cardinality. Homework Equations The Attempt at a Solution I have to prove bijection. The injective part is obvious. Say, A =(1,3) and B =[1,4] f: A → B f(x) = x It's...
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    GCD of four Different Pairs

    Homework Statement Give an example of a set S of four (distinct) positive integers such that the greatest common divisor of all six pairs of elements of S is 6. Homework Equations The Attempt at a Solution Can I say that my numbers are in the form? 6 12 18 30 Is this ok?
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    Set Theory - Uncountable Sets

    Set Theory -- Uncountable Sets Homework Statement Prove or disprove. There is no set A such that ##2^A## is denumberable. The Attempt at a Solution A set is denumerable if ##|A| = |N|## My book shows that the statement is true. If A is denumerable, then since ##|2^A| > |A|, 2^A ##...
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    Cardinalities of Sets

    Homework Statement Explain why ## (0,1 ) ## and ## (0,2)## I have proved that a function ## f: (0,1 ) -> (0,2) ##defined by f(x) =2x is bijective. Homework Equations The Attempt at a Solution I could state that due to the sets relationship being bijective they have the same cardinality...
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    Definition of Funtion

    Homework Statement Let ## C= \{ x \in R : x \geq 1 \} ## and ## D = R^+ ## For each f defined below, determine ## f(C), f^{-1}(C), f^{-1}(D), f^{-1} (\{1\}) ## a.) ## f: R -> R ## is defined by ## f(x) =x^2## I have problems with the definitions The Attempt at a Solution a.) ## f(C)= { 1 ...
  20. K

    Given the Set S ={1,2,3,4}. Define a relation on S that

    Homework Statement a.) Is symmetric and transitive, but not reflective: b.) consists of exactly 8 ordered pairs and is symmetric and transitive: The Attempt at a Solution If the question asks me to define some relation, do I need to define some math property like power of some number or...
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    Equivalence Relations problem

    Homework Statement Let ## H = \{ 2^{m} : m \in Z\}## A relation R defined in ##Q^{+} ## by ##aRb ##, if ## \frac{a}{b} \in H## a.) Show that R is an equivalence Relation b.) Describe the elements in the equivalence class [3]. The Attempt at a Solution For part a, I think I am able to solve...
  22. K

    Prove Existence Unique Real Solution

    Homework Statement Prove Existence Unique Real Solution to ## x^{3} + x^{2} -1 =0 ## between ## x= \frac{2}{3} \text{and} x=1## The Attempt at a Solution ## x^{2} ( x+1) =1 ## I know that the solution is x =0.75488, but this came from some website. How do I find this number without a...
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    Congruency and Divisibility of Odd in Z

    Homework Statement Let n be any odd integer. Prove that 1 is the only "common" divisor of the integers n and n+2. The Attempt at a Solution I don't think I understand the question. The few notes I have state d| (n+2 )- n This resembles n+2 ##\equiv## n mod d , but I don't see the...
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    Congruence Modulo Proof

    Homework Statement Suppose n is an integer which is not a divisor of 5. Prove that ##n^{4} \equiv 1 mod5## The Attempt at a Solution I know that 16 mod 5 is equivalent to 1 mod5. ##16 = 2^{4}## 2 is not a divisor of 5. How do I prove this for the general case? I know ##n^{4} -1...
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    Prove Absolute Value Property

    Homework Statement Prove that for every two real numbers x and y ##|x+y| \leq |x| + |y| ## Homework Equations The Attempt at a Solution There are three cases. The easiest ones is when they are both positive and negative. The third one I have problems with. The numbers have different...
  26. K

    Trivial and Vacuous Proofs

    Homework Statement Let ##n\in N ##. Prove that if ##|n-1|+|n+1|\leq 1,## then ## |n^{2}-1|\leq 4## Homework Equations The Attempt at a Solution I am trying to show by a counter example that this statement is not true. Consider this statement: ##|n-1|+|n+1| \leq 1## Assume ...
  27. K

    Set Theory Logic Question

    Homework Statement In each of the two following open sentences P(x) and Q(x) over a domain S are given. Determine all ##x \in S## for which P(x) → Q(x) is a true statement. ## P(x): x \in [-1, 2]; Q(x): x^{2} \leq 2; S=[-1,1] ## Homework Equations According to truth values for →: a...
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    Determine Union of Sets Belonging to Interval

    Let ##I## denote the interval ## [0, \infty )## . For each r ## \in I ## define: ##A_{r} = \{ (x,y), \in ##R x R : ## x^{2} +y^{2} = r^{2} \}## ##B_{r} = \{ (x,y), \in ##R x R : ## x^{2} +y^{2} \leq r^{2} \}## ##C_{r} = \{ ## ... ## : ... > r^{2} \} ## a.)...
  29. K

    Deterministic Finite Automata - Myhill Nerode

    Homework Statement Let ## f\left( b_{1}, \dots , b_{n} \right) ##be a boolean function. Define ##S_{f} = \{\left( b_{1}, \dots , b_{n} \right): f\left( b_{1}, \dots , b_{n} \right)=1; b_{i} \in \{0,1\}, 1\leq i \leq n \}## The subsets ##S_{f}## are viewed below as languages consisting of...
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    Bi-variate Non-Homogeneous Polynomial Conceptual Question

    Homework Statement I have found the roots of my polynomial: ## (2x+3y)^{2}-1 =0 ## Roots are x=3n+2 & y=-2n-1, where n belongs to all Z. What does it mean that the solution has arbitrary large coordinates? The Attempt at a Solution I think I know the basic concept of root. It could be...
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