1. Let A = {z € Z:z = 2 (mod 5)}, and B = {y E N : y² < 90}. Compute the sets ANB, and B\ A in roster notation.

Q: Give an enumeration procedure for the set L = {ab*a™: n> 1, m >1}.

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Q: (a, o) is not measurable.

A: to find that a,∞ is measurable or not  proceed as follows We know that every interval is measurable.…

Q: &2/ suppost se =20, 1} and c={208{ fie ld =20,18 and e= Enumerate the class of all 6- Containing C…

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Q: For two R-modules Mand N Show that Hom (M,N) any with the composition is not agroup?

A: Given that M and N are two R-modules. We have to show that HomM,N is not a group under composition.…

Q: 5. Prove that S(n +1, k+1) = £ $(m, k) (").

A: To prove- Sn+1,k+1=∑m=knS(m,k)nm The falling factorial is defined as the polynomial,…

Q: 7. Show that if A and B are similar, then det A = det B. %3D

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Q: 6. Prove that if alb and k E Z, then a|(b+ ak).

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Q: Show that if m e Z+ and a e Z such that gcd(a, m) = 1 and ordma = m – 1, then m is prime.

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Q: Consider the Venn diagram shown below. A В 8 12 5 10 3 C 2 Determine the following cardinalities.…

A: n(A) = content of An(A) = 8+5+1+7n(A) = 21

Q: 2. Suppose a, b, and care integers. Prove that if a |band alethen a |b+c and a|b-c.

A: Assume a|b and a|c.

Q: Let A (k,s), t.en the power set of A is:

A: Set of all subsets of A is called power set of A.

Q: 4 Show that a + B = B +a for all a, BE C.

A: We can write alpha and Beta in general form of element of complex number.

Q: SUppose K,n EZ and O < Kcn. Prove +hat by (onsi nsi dering subsets of (1,... --- --- (^) - (":') + (…

A: The given problem is to prove the following identity using the basic mathematical identities.

Q: If m, n, p, q are integers, n > 0, q > 0, and r = m/n = p/q, prove that

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Q: Theorem 1.4.1 Let n(A) denote the number of elements in set 4 n(A, × A2 × ...× Aµ) = n(A) n(A2).…

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Q: I, For A and B of Hhe univer sal ß subsets proside a in terse ct B set X, provide a formal prof for:…

A: Given: X is a universal Set. A and B are the subsets of Universal Set X. We have to prove that A…

Q: Prove that if n 2 3, the only element o of S, satisfying oy =ya for all y E S, is the identity…

A: Let n≥3. Let σ∈Sn such that σγ=γσ for all γ∈Sn. Suppose σ≠i, the identity Then there exists…

Q: Prove that If k, q, w and r are integers, and if c|k and c|q, then c|(kw + qr)

A: We know that  Definition of Divisibility : If a and b are integers such that a≠0, then we say "a…

Q: 2. Compute the convolution of the following signals for n 20 n n x1(n) = and x2 (n) = (;)" u(n) n…

A: Consider the expression of . Determine the z-transform of .

Q: In AAGE and AOLD, ZGAE = ZLOD, and AE OD. To prove that AAGE and AOLD are congruent by SAS, what…

A: Given Data: Congruent angles:∠GAE≅∠LOD Congruent sides: AE≅OD The triangles △AGE and △OLD are…

Q: Consider the following sets: A = {x €NIX >0 AxS 100 A x mod 2 = 0 } B = {X€NIX>0Axs 100 A x mod 4 =…

A: “Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: Suppose A1, A2, ..., 430 are thirty sets each having 5 elements and B₁, B2, ..., B₁ are n sets each…

A:

Q: Derive k, kkg and ky fourth order Runge Kutta methed-

A: Range Kutta

Q: Suppose a tair Coin is flipped twice, het A, = "Head on 1st Toss" A= "Hend on 2 Toss" Ag=" %3D 1)…

A: Suppose a fair coin is flipped twice, let A1=”Head on 1st toss”, A2=”Head on 2nd toss” and A3=…

Q: For all s, Isin s≤ 1

A:

Q: Let {0, 1}N = {(a, a1, a2, . ..) : an = 0 or 1, n E N}. • Construct a 1-1 correspondence between {0,…

A: let 0,1N=a0,a1,a2,....:an=0 or 1, n∈ℕ (i) construct a 1-1 correspondence between 0,1ℕ and all the…

Q: 1. Let x, y, z be integers. Prove that if x(y + z) is odd, then x is odd and at least one of y or z…

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Q: U is a neighborhood of x0 and U ⊂ V, The V is a neighborhood of x0. (2) If U1 ...., U2 are…

A: Solution ::

Q: Answer the following questions about a deli system in a grocery store. Within the system, every…

A: (a) The customer IDs are ranging from p to q  inclusive. and meats with IDs ranging from r to s,…

Q: There are two food carts serving food at a local park. At any given time, Let A denote the number of…

A: Hello! As you have posted more than 3 sub parts, we are answering the first 3 sub-parts.  In case…

Q: 1. Suppose A, B, and C are sets, A. Prove that: AX (Bn C) = (A X B) n (A x C) %3D

A: As per our guidelines we are supposed to answer only one question.Since you have asked multiple…

Q: (b) Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7. Deduce that…

A: Given X, Y and Z are subsets of 1, 2, 3,.......,10. Cardinality X=Y=Z=7. To prove that X ∩ Y ∩ Z is…

Q: Let d, a, b E N. Prove that

A: Given,  d, a, b∈ℕ To prove that if d|a+b2  and   d|a-b2, then d|b.

Q: 2) Let mf lu and (a,m)=| a,bE Z such that Let h= Om ( a) show and and (h,m)=1, %3D let. ne OmCab)…

A:

Q: Let (a) H = {(), (1,2), (3 4 5 6), (35) 46), (36 54), (12) ( 3 4 5 6) (12) 35)(46), (12) (3654) 3 K…

A:

Q: Suppose that X is a subset of Y, such that P(Y)>0. Is true or false that P(X|Y) is equal to…

A: It is given that X is a subset of Y This means that PX∩Y=PX...(i) The formula for PX|Y is given by:…

Q: Let A, B and C belong to M(3,R) such that: det(A) = 10, det(B)= 10/5 and det(C) = 3, then…

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Q: 2. 20 Wnat is the loargest order of an element in Zs Ze O Z 2) Exhibit an element with this arder,

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Q: 12.42. Let a, be Z, where not both a and b are 0. Prove that if d = gcd(a, b), a = ajd and b = b1d,…

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Q: 2. Let S be the set [9]: (1.2.9). (c) List 5 derangements of S having 2 in the 3rd position and 5 in…

A:

Q: State whether the given set E is bounded or unbounded and justify your answer with a proof. E = {z €…

A: We are given the set, E = { z ∈ Z : z ≡ 7 mod 9}. First, we will try to find the elements in E then…

Q: 1. Given a deck of 30 cards labelled with numbers 1- 10 (with 3 cards of each number), how many…

A: From the given information,  A deck of 30 cards labelled with numbers 1-10 (with 3 cards of each…

Q: Let Σ = {0, 1, +, =} and S = {x = y+z : x,y,z ∈ {0,1}* and x,y,z are binary integers, and x is the…

A: Given that S=x=y+z By using the pumping lemma. Assume that for the purpose of reaching a…

Q: Define the sets D = In E N} n+1 Prove whether D is finite, countable, or uncountable?

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Q: let A to have0 al an must by

A:

Q: 4.1 Consider the three permutations in S6: 1 2 3 4 5 6 4 5 6 2) 3 1 1 2 3 4 5 136 5 2 4 1 2 3 4 5 S…

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Q: Suppose that a,beG such that a = m and b =n with Gcd (m,n) = 1. Then n | is in mn

A: Suppose that a, b∈G such that a=mb=n and Gcdm, n=1 i.e., m and n are relatively prime. We will find…

Q: Let A= then det(3A") = %3D O 18

A: We use Matrix and determinant concept here.

Q: Show that 1 (N, <) is well-ordered. 1 2 (Z, <) is not well-ordered. 3 (R, <) is not well-ordered.

A: Please send questions as different questions. We cannot do more than 3 at a time.   A relation ≤ on…

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.