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Q: Prove that the set {a E R: a2 not equal to 6} is open.

A:

Q: Show that if f : A → B and g : B → C are onto, then gof : A → C is also onto.

A: Given, f : A → B and g : B → C are onto.

Q: C. How many extensions of g to A are onto?

A:

Q: Prove that Vn e N (the set of natural numbers), ÷+ ɛ N. 3 +

A: We have to use proper manipulation.

Q: Consider the set T = {..., −34, −17, 0, 17, 34, ... } with the usual definition of less than (a0).…

A: Completeness property of real numbers:  Every non-empty set of real numbers that has an upper bound…

Q: An element x is in A ∪ B if, and only if, _____________________.

A: Known fact: A∪B=x | x∈A or x∈B

Q: Let C: z-3 and I – 6dz. ONone of these

A:

Q: Let ficxT)→(メッテ)and g: メっテ)→(Kっズ) e are homomorphism then gof 1s homo.

A:

Q: Prove that the class {AC – B, BC – A, AB − C} is pairwise disjoint.

A:

Q: Let a and ß belong to S,. Prove that a-'ß-laß is an even permutation. n°

A:

Q: Stefan was able to map AABD onto AABC. D A

A: Given, ∆ABD  and ∆ABC     We have to find whether we can map  ∆ABD  onto ∆ABC

Q: norn rmed M Y is closed if and only if the

A: Given result: Let X, Y be the normed linear spaces  M subspace of X To show: T: M →Y is closed if…

Q: d. a specific element of R° that is not in V. е. a specific element of v in V for which -v is not in…

A: Given (a,b,c) where c=a-2b

Q: Jin Let G beagreupsuch that I6tefs thens

A: Answer

Q: Given d, = and dy = Then d, and d, are not - conjugateif Non of these O D

A: Given d0 = -11  and  d1 = 56

Q: For all a be a, a=gal cabl id and only it acdinde b

A: We have to prove that for all a, b∈Z+, a=gcda,b if and only if a divides b. First we will prove if a…

Q: Let a and b belong to Q with b ≠ 0. Show that Q(√a) = Q(√b) if and only if there exists some c ∈ Q…

A:

Q: Prove that Z ≈ E*, where Z is the set of integers and E* is the set of positive even integers.

A: To prove:

Q: Give an example of a closed subset S of R^2 such that conv S is not closed

A: An example of a closed subset S of R2 such that conv S is not closed is given below.

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

A:

Q: Let 9={a: aER}, then T is Borel algebra.

A: Given a statement

Q: Let G=(V, E) be a finite

A:

Q: Show that gcd(a,n)=1 if these exist m and k such that ma+nk = 1.

A:

Q: Let a, be such that asb. Prove that tx Eqt ax cbx

A: Concept: We have a,b∈Q means that a, and b belongs to the rational numbers and a&lt;b means that a…

Q: .

A:

Q: (a) Prove that A° is open, A° C A, and if U CA is open then UC A°. (b) Prove that A is open if and…

A: Since it is not mentioned which part answer we are supposed to do, so we are doing first part.…

Q: 4. Prove or provide a counter example for the following: (AU B)nC = (AnC)U(BnC)

A: We need to prove that(A∪B)∩C=(A∩C)∪(B∩C)If we have to prove that(AUB)∩ C C C…

Q: 2. Prove that any two non-zero elements a, b in a P.I.D. R have a g.c.d. Further if d e R is a…

A:

Q: Prove or disprove that if (A U B) C (An B), then A = B

A:

Q: Given that * :Zx Z Z is defined by m * n = m" for m, n e Z, then is an operation on the set of…

A:

Q: an be cauchy, prove why a subsequence of (an) must also be cauchy.

A:

Q: o Let A-

A: The given matrix is : A=1-111 Calculate the value of norm A∞. We know the formula of the infinity…

Q: If A and C are nonempty classes, prove that AC B and C C D if and only if AxC C BxD.

A:

Q: Let A be uncountable and B be countable. Prove that A\B is uncountable.

A:

Q: Let ü = 21 – k and i = i + 3j + k. Compute |ū], |ū], ū × ĩ, and Projzu.

A:

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: Let α, β, γ ∈ Z. Prove: If  α2 + β2 = γ2. then α or β is even.

A:

Q: Prove that if A C B then P(A) C P(B).

A: Proof:

Q: Prove that there is no permutation a such that a'(12) a = (34) (15). %3D

A:

Q: If A⋂B are independent, prove that Ac⋂B are independent.

A:

Q: Let a * b = 2ab be defined on Z*. 1. Is * Commutative? 2. Is * Associative? 3. What is the identity…

A:

Q: Let A = {a,b, c} and B = {b, c, a}. Prove that A = B. Thus, the order %3| in which the elements are…

A: The given sets are A={a,b,c} and B={b,c,a}we have to show that A=BResult : two sets A and B are…

Q: Let 9-(a: a€R}, then T is Borel algebra.

A: Clearly the measure of the set is =0 .

Q: (b) Is T one-to-one, onto or neither.

A:

Q: Prove that if A,B ∈Mn×n(F) are similar, then det(A) = det(B).

A: If the matrices A and B are similar then there exists a non singular matrix S,

Q: Let Rt st a a, b€ Z and let D=R a-4. 9. show that o E homomosphic Romemerphic Momor

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Q: If A subut of B. where B 18 a closo d subset of IR show that

A: Given: A is a subset of B, where B is closed subset of ℝp. To show: A¯⊆B.