Let n > 2 be an integer and k e Z. Denote by [x] the congruence class of x € Z modulo n. Show that either there exists l E Z such that [k]-[1] = [1], or there exists a congruence class [1] # [0] such that [k] · [l] = [0]. Prove as well that one possibility excludes the other.

Q: Suppose gcd(a, n) = 1. Show that there exists b such that [a]n * [b]n = [1]n.

A: Using division algorithm.

Q: Let ℤn denotes the integers (mod n), e.g., ℤ10 denotes the 10 equivalence classes ℤ10 =…

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Q: 26. Let E be the set of all even integers, and let O be the set of all odd integers. Let X = = {n =…

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Q: 4. Let F : A --> B, g : B --> C, and h : B --> C be functions, and suppose that f is onto. Prove or…

A: Given: f:A→B, g:B→C and h:B→C are the functions and f is onto.

Q: Suppose K is an extension of F of degree n. Prove that K can bewritten in the form F(x1, x2, . . . ,…

A: Suppose K is an extension of F of degree n. To Prove: that K can be written in the form F(x1, x2, .…

Q: Consider E={-2+1/n} n=1 to infinity U (3,9) as a subset of R with the usual definition of < (less…

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Q: 5*. Suppose that 0: G -+ H is a homomorphism. Prove that (a) (a")= (x)" for all positive integers n.…

A: To prove the properties of group homomorphisms

Q: Consider the following argument for showing that |R| = |P(N)|. By the Cantor-Bernstein-Schröder…

A: As per the question, we will be verifying the argument ℝ=Pℕ. Let us summarize the given information.…

Q: For each statement below decide if it is TRUE or FALSE. • There are two integers n and m such that…

A: We will find out whether the given statements are true or false.

Q: Let n = 2k with k ≥ 2. (a) Prove that pk is an element of order 2 which commutes with all elements…

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Q: Let m ∈ Z and consider the relation Rm on Z given by aRmb if m|a − b. (a) Show that Rm = R−m. (b)…

A: Given Let m ∈ Z and consider the relation Rm on Z given by aRmb if ma-b(a) Show that Rm = R-m(b)…

Q: n Exercise 2.11. Prove that n = [2] + 10 for all n E Z.

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Q: {(a, b) ab >0} | {(a, {(a,b) | a² > b} ¯ {(a,b) | a² = b² (mod 7)} b)|a>b-1}

A:

Q: Define a function S : Z+ → Z+  as follows. For each positive integer n, S(n) = the sum of the…

A: Define a function S : Z+ → Z+  as follows. For each positive integer n, S(n) = the sum of the…

Q: Prove that if x is not free in the formula A, then ` A ≡ (∃x)A

A: The formula is -------(1). To Prove: x is not free in the formula Proof: Let us assume that A(x) :…

Q: Let n > 1 be an integer. We say that [x] ∈ℤ/n is a zero divisor if there exists [y] ∈ ℤ/n − {[0]}…

A: Let n&gt;1 be an integer.  Given details, [x]∈ℤ/n is a zero divisor if there exists [y]∈ℤ/n-{[0]}…

Q: 6. Given the following relations on the set of all integers where (z,y) = R if and only if the…

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Q: Suppose n1,n2 are positive integers and gcd(n1,n2) = 1. Then for every integer x, if n1 | x and n2 |…

A: Given that gcd(n1,n2)=1 Then there exist integers m1 and m2 such that          m1n1+m2n2=1 Then…

Q: Let A = {1, 2, 3, 4} and consider the relation R on A given by aRb if and only if a−b ≡ 0 mod 4, or…

A: Given: The set A=1, 2, 3, 4, and the relation R:A→A as  R=a,b: a-b≡0mod4 or a-b≡1mod4 or a-b≡-1mod4…

Q: 2. List all the polynomials in Z3[x] that have degree 2.

A: To find all the polynomials in Z3[x] that have degree 2

Q: Prove or disprove the relation C on Z defined by C = {(x, y): x=y (mod 5)} satisfies the two…

A: I just used the definition of congruence and divisibility

Q: 6. Is the function h: Z → Z defined by ( 2n if n 20 h(n) -n if n < 0 one-to-one? Is it onto?

A: The solution is given as follows

Q: 5. Find all integers r with 0 < a < 10 satisfying the following congruence relations. If no such r…

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Q: [10] 1. Prove that that the natural numbers with the binary operation of multiplication (as defined…

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Q: Prove that if n > 2 then φ(n) is even

A: The mathematical function known as Euler's totient function, or φ(n), counts the number of positive…

Q: (a) x + y = 0 r+ y= 0 A. symmetric B. transitive C. reflexive D. antisymmetric (b) xy > 1

A: