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 – {[0]} such that [x][y] = [0]. We say that [æ] is a unit if there exists [y] E Z /n such that [x][y] = [1]. 7a) What are all the units in Z/12? What are all the zero divisors? b) What are all the units in Z/14? What are all the zero divisors? c) What are all the units in Z/13? What are all the zero divisors? (You do not need to prove your answers to this question, but you should show your work.)

Q: Let i be a positive integer and A, [-i+ 1,1]. Then U14, %3D O R O None of these O F1,11

A: Given that 'i ' be a positive integer and Ai=-i+1,1. We have to find the union ∪i=1∞ Ai.

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Q: Let a, b c and d be real numbers. Show that ab-cd = b(a-c)+c(b-d), and deduce that |ab - cd| ≤…

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Q: Give a direct proof of the theorem: “if x and y are odd integers, then x + y is even.”  DISCRETE…

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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: Let a, b, x, y ∈Zbe such that ax+by= 11. May we conclude that gcd(a, b) = 11?Why or why not? (Write…

A: Consider the given information:Let a,b,x,y∈ℤ be such that ax+ by=11.

Q: Simplify Z = [(PNR) u ā] n (PU Q) n P - R using set identities Q]

A: Set Identities:-Commutative law:- A∩B=B∩A  and  A∪B=B∪AAssociative law:-   A∩(B∩C)=(A∩B)∩C  and…

Q: 4. (а) Let a, b, c E Z. Prove that, if a | b, then ac | bc, but that the converse may be false. (b)…

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Q: For any two non-negative integers x and y, let d(x, y) = x + y. Let x, y, z be any three…

A: See solution below

Q: Let x, y, z, w ∈ Z and n be a positive integer. Suppose that x ≡n y and z ≡n w. Show that x + z ≡n y…

A: Suppose that x ≡n y and z ≡n w then, (i) we are assumed that  x-y=k1 n  and  z-w=k2 n  for some…

Q: 5. Prove the following identity for n 2 1: Fzn+1 = F+1+ F;

A: We need to prove that ,  F2n + 1 = F2n+1 + F2n We know that ,   Fm+n = FmFn+1 + Fm-1Fn ,  m ≥ 1 , n…

Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

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Q: Show that for integers a, n satisfying 0 < a <n, ged(a, n) = gcd(n, n-a)

A: Solution is given below:

Q: Let y1, . ... Yn real numbers, and let n y = 1/√ Σ² ²=1 Yi n their arithmetic mean. Show that for…

A: The proof is explained in the explanation section.Explanation:Solution: Let's prove the given…

Q: Let a E R which expressions are correct? and give the justification why they are part of the Borel o…

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Q: Show that a,b,c are in A.P. If x is the G M be tween a and b and y is the G M between b and c,  then…

A: We have to show if  a, b, c are in A.P.,  x is the G M be tween a and b and y is the G M between b…

Q: ∃x ∈ D such that ∀y ∈ E, x + y = 1 Negation:

A: We need to find the negation of  ∃x ∈ D such that ∀y ∈ E, x + y=1

Q: Let a ER which expressions are correct? and give the justification why they are part of the Borel…

A: Consider the given information: Let a∈ℝ  Then which statements are correct. Here Second option is…

Q: For allne z, gcdin,nt)=l.

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Q: 3. A = {2, 2+2, 2+4,2+6, 2+8} where A = {2,4,6,8} 4. A=(1-22, 1+22) where A = Z*

A: Note : (i) ∩α∈S Aα=x : ∀α∈S ,x ∈Aα               (ii) ∪α∈S Aα=x : ∃α∈S ,x ∈Aα

Q: Determine whether the following generalizations are true or false. 1.  If x and y are both elements…

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Q: (9) Prove that for integers x, y and z, if x² + y² = z² then either x or y is even.

A: We use method of contradiction to prove given statement.

Q: 2. Let f and g be any permutations. Show that the order of h = fgf- is the same as the order of g.

A: Here f and g are tow permutations  and we are given that h=fgf-1. We have to show that both f and g…

Q: Is 1z = 1 for all z ∈ C?

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Q: 6.3.6 For each real number x, let Ay = {3, –2} U {y E R : y > x}. Find U Ax and N Ax. xER xER

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Q: Find the characteristic equation of the recursive relation an 송,Mn 2 1. O 3r – 1=0 Or+3=0 Or-3= 0…

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Q: let G be the set of all triples of the form (k1, k2,1) or (k1,k2,-1), where the ki, i=1,2 are…

A: Given :  G = k1, k2, ±1 : k1, k2 ∈ℤ with the operation  k1, k2, 1l1, l2, -1 = k1+l2, k2+l1, -1 To…

Q: Let E-f(): 05 05 n. 05 25 1.7√/25 rs 7). Compute 6.8365-1051090 11²³ dv

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Q: . Let x be a positive real number and d be a positive integer. Prove that the number of positive…

A: [.] is the greatest integer function.

Q: (1) if is reflexive, symmetric, or transitive. For a, b = Z, define ab if a + b is a multiple of 3.…

A: A relation is a subset of the cartesian product A×B where A and B are non-empty sets. For cartesian…

Q: 1 1 1 (c) Find simple descriptions for each of the sets U • U [ ₁ - ² x + ¹) and № [² - ² + ¹) X- n…

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Q: 1.5 Prove that e-2=| 0.

A: