1. Find all cquivalent classes determined by congruence modulo 5 on Z.

Q: Construct the Addition Table of Congruence Class Z5.

A: The number of elements of the given group is written as: ℤ5=0,1,2,3,46=1=−45=0,6=1,7=2 and 8=3and if…

Q: 5. Show that M:= {ECR: E is countable or E is countable} is a o-algebra.

A:

Q: Solve the congruence 2x + 3 = 11 (mod 7). (Select all congruence classes below (modulo 7) that…

A:

Q: 2. Evaluate the lIntegral viing any a ppopriate algebre ic metad ir igohometry identity

A: To Evaluate: ∫2t2+t2+1t2+1dt Concept: i. ∫afxdx=a∫fxdxii. ∫f+gdx=∫fdx+∫gdxiii. ∫11+x2dx=tan-1xiv.…

Q: Example 2 Using the SAS Congruence Postulate Prove that ASYT = AWYX. Statements Reasons 1. TY = SY =…

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Q: 6. (i) Solve the congruence relation 3x + 5 = 2(3 – 5x) (mod 35) for r. - (ii) If a = b (mod n) and…

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Q: 13. Is (Sn,) a commutative(abelain)? Is (Sn,) cyclic?

A: Solution is given as below :

Q: 2. List all the subgroups of U(15) under multiplication mod 15 and draw the lattice diggram,

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: 1. Use Venn diagrams to exhibit A - (BnC)=(A – B)U (A – C). 2 Use memberchin tables to prove that4…

A:

Q: Recall that tn denotes the nth triangular number. Prove the following identity: tn+2k = tn (mod k)

A: If tn is the nth triangular number, then tn=nn+12

Q: The linear congruence ax = b(mod m) has a unique solutionif and only if: gcd (a,m) =1 gcd(a,m) =1 o…

A: First option is correct. gcd(a, m)=1.

Q: . Prove the following: If gcd(a, b) = 1 and ca then gcd(b, c) = 1

A: We will prove the given statement.

Q: Theorem 3: Ifconverges to l, ther any subsequence ofalso comverges to l.

A: We have given that sn converges to l. Let snk be any subsequence of sn. We need to prove that snk…

Q: Prove that the number of solutions to the congruence xk ≡ 1 mod p is gcd(k, p − 1)

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Q: Show that for every x E IR, there exists an n EN such that <x<n.

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Q: Prove that An B = AUB using the method of i) membership table, ii) identities, and iii) Venn…

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Q: 1. Let U24 = {z E C|224 = 1}. Define £24 = cos () + i sin (5). (a) List all subgroups of U24 and…

A: Given:  U24=Z∈C and z24=1 and ξ24=cosπ12+isinπ12 We know that cos x+i sin x=eix So,…

Q: 2, Say whether the Runction is even, odd, or neither. Give reasanS for your dnswer. ga) = x"+ x*-1

A:

Q: 1. List all the subgroups of Z6 under addition mod 6 and draw the lattice diagram.

A: ℤ6={0,1,2,3,4,5} ℤ6 is a cyclic group Divisors of 6=1,2,3,6

Q: Find the sum of the GS that satisfies the given Condihon n- 4 az=30 a3S

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Q: Let H x K = { (h,k) |h in H, k in K} such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A: here ℤ2=0,1

Q: The linear congruence ax = b(mod m) has a unique solutionif and onlyif: gcd(a,b)= 1 gcd(a,m) #1 O…

A: Gcd(a,m)=1

Q: 1. Show that 2465 is a Carmichael number.

A:

Q: Which of the following structure is not a binary structure? إختر أحد التبارات .a O (@'.+) (0', -) **…

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Q: 8. Find a for and az geometric sequ E5103 and the 3D21 given

A: consider the geometric series   a , ar , ar2, ar3, . . .  where a1=a

Q: Use symmetry to evaluate the followingintegrals.

A: The integral is ∫-π/4π/4tanθ dθ.   Check if the function is odd or even. Substitute -θ for θ in the…

Q: Theorem 24. If ɛ > 0, then there is an integer N EN such that 0 < + < e. .

A:

Q: Suppose that lal = 12. The number of distinct left cosets of in is: * %3D 12 O 10 O None of the…

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Q: O lim Sop of Sn= 61)" (24 ) is ?

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Q: Suppose that " is in lowest terms and that m< n. Then has a simple-periodic dec- imal representation…

A: Introduction: The ratio of given two numbers could be represented as a factor less than one always…

Q: 5. (a) Ifp is a prime then the group U, has (d) elements of order d for each d dividing p- 1. (b)…

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Q: How would we show that H is cyclic?

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Q: (8) Let a and b be reel mumbers such that a>o and bo and beD is not enough to gire a uniform formula…

A: No option is correct. Correct answer will be: - arctan(b/a)

Q: Consider the subring S = 2216 of Z16. Then char (S) = 4 6. 8. оОО О

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Q: 3 S how that Set M with n elements. a where n is a elem ent of ne tural numbers (ne IN. has 2…

A: Number of elements in M is n. To prove power set of M has 2n elements. We know that Power set of M…

Q: List all the subgroups of U(15) under multiplication mod 15 and draw the lattice diagram.

A:

Q: Show by means of a membership table that both absorption laws are identities.

A: According to the given information, it is required to show that the absorption laws are identities…

Q: The linear congruence ax = b(mod m) has a unique solutionif and only if: gcd(a,b) 1 gcd(a,b) = 1…

A: gcd(a,m)=1

Q: 10. Using the Chinese Reminder Theorem, compute the value of x € Z+ which satisfies the following…

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Q: (4) Let A CRbe a et ith greatet element a E A Prove that sup(A)=a

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: If 2%= N₁, then a Ramsey ultrafilter exists.

A:

Q: 6. Describe the left cosets of SL2(R) in GL2(R). What is the index of SL2(R) in GL2(R)?

A: For a group G with a subgroup H, the left coset is defined by gH=gh:h∈H∀g∈G.  Since g can be…

Q: Use the result in Problem 3 and state conditions on n, a, and k under which the congruence * a (mod…

A: For (4), Here, we need to find conditions on n, a, and k under which the congruence xk≡a mod n is…

Q: 1. Determine if each of the linear congruences below is solvable. If a congruence is solvable, then…

A: The general form is ax≡b(mod m), first check is this linear congruence solvable or not? for that, we…

Q: If p,q,r are prime number but p is not equal to q, then what is gcd(p,q)? What is the smallest…

A: It is given that p and q are prime numbers. Assume that gcdp, q=d. Then, d is the greatest common…

Q: Is there any n n for which U2n = Dn?

A:

Q: Prove that ANB = AU B by giving an element table (also known as a table) proof. membership

A: If two expressions are equivalent, then the values in their respective columns in the membership…

Q: 2.1 Is it a group? For every question write down if a group is Abelian. 1. Does the (Z, +) (Integers…

A: According to our guidelines we can answer only three sub-parts and rest can be reposted.

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.