. Suppose that p is a prime integer. (a) Show that if [x], [y] € Z and neither of which is [0], then [x] · [y] ‡ [0]. . (b) Show that for every [x] € Zp, [x] ‡ [0], there exists a [y] so that [x] · [y] = [1].

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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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A: This is a question on the properties of continuous function.

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: -. For any integer not an integer, then [x]+[m- x] = m-1. m and any real number x, if x is

A: Let m be any integer and x be any real number that is not an integer.

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Q: 1. For an integer k, let [k] = {1, 2, 3, ... , k} denote the set of the first k integers. Let Fr be…

A: Given for an integer k, let [k]={1,2,3,...,k} denote the set of the first k integers.

Q: Prove this is an equivalece relation. x related to y iff x2 -  y2 is even

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Q: Suppose X and Y have joint characteristic function (s, t). Show that, subject to the appropriate…

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Q: Let n > 1 be an integer. We say that [x] E Z/n is a zero divisor if there ex- ists [y] E Z /n –…

A: Introduction: The units in a ring is very important. It is not necessary that every ring has…

Q: Let a be a non-zero element of Z43. Explain what it means for |a] to be of order w, and hence prove…

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Q: Let f(x) = a,x" + a, p* + ...+ a, E Z[x] and a, + 0. Prove that if r and s are relatively prime…

A: It is given that,  f(x) = anxn + an-1xn-1 +...+ a0 ∈ ℤx  with an ≠ 0. Also, it is given that r and s…

Q: 1. Give an example of a function f N N which is injective but not surjective.

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Q: 4. Consider the following relations R on the integers Z. For each of these relations, say whether…

A: For the solution of the problem follow the next steps.

Q: 2.a) Ση-0(- 1)" χ" = 1+x for |x| < 1

A: This is a question of real analysis. L.H.S is a geometric series with common ratio (-x)

Q: Let the p.d.f. f(x) be positive on and only on the nonnegative integers .Give that f(x)=4/x f(x-1),…

A: Given that:

Q: Let f(x) = axm + am-prm-1 + . . . + a, and g(x) = b,x"+ b,-px"-1 + ... + bo belong to Q[x] and…

A: Given that, f(x)=amxm+am-1xm+....+a0∈Q[x]g(x)=bnxn+bn-1xn-1+....+b0∈Q[x] This implies that ai,bi∈Q.…

Q: Prove that P[X |Y ] = (P[Y | X] · P[X] ) / P[Y]

A: From the definition of conditional probability, for two events X when Y is given PX|Y=PX∩YPY

Q: Let S be the set of all positive integers. Define: S -> S by f(1) = 2,  f(2 ) = 3, f(3 ) = 1, and…

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Q: ind the m<t

A: Given: m&lt;R=140 m&lt;S=80 Find: m&lt;T=?

Q: prove for any 2 non negative integers x and y, where x>y, if their gcd IS Y, THEN THEIR LCM is x

A: The objective is to show that LCM(x,y)=x, given GCD(x,y)=y; where x and y are non-negative integers…

Q: 2. Let ECR. Show that E is Lebesgue measurable if and only if there exists a sequence of open sets…

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Q: Let n >= 3 be an integer. Let Vn(x) = {[x 1 1 1 ... 1 1]T,[1 x 1 1 ... 1 1]T, [1 1 x 1 ... 1 1]T,…

A: By the consequences of vector spaces we solve the given problem.

Q: Prove that if x is a real number such that x ≥ 0 and x ≤ € for all € E R where € > 0, then x = = 0.

A: This question is from real analysis.Answer given below.

Q: Show that for any real numbers X and y. one of the following two case can happen: Either [x+Y = [×]…

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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: How many functions f: {1,2,3,4} → a. ƒ(1) = 1 or ƒ(2) = 2 (or both)? → {1, 2, 3, 4} are there…

A: Since your question carry more than three subparts , so as per Bartleby guidelines we are answering…

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

A: Given: K is an extension of F of degree n.We need to show that K=F(x1,x2,...,xn) for some…

Q: 4. Prove that if (an) is increasing and an ̸= 0 for all n ∈ N then ? 1/ an is decreasing.

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Q: 2. Identify which of the following subsets of R are closed, open, or neither. a. (zz is an integer }…

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Q: Let a₁ ≥ a2 ≥ ….. ≥ an and b₁,b2, ..., bn a1 b real numbers such that for each k, 1 ≤ k it ant

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