The topic is fields and boundedness. Statement: Given that sup(S) is a finite number, and we assume that there is a positive "y". Prove: There exists an element x  in S such that α - y < x. PS: α = sup(S)

Q: Prove that S = subring of R[x]. {p(x) = R[x] | p(2) = 0 or p(3) = 0} is not

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Q: a) Prove that P(EUF) ≤ P(E) + P(F).

A: Given that Prove that P(E U F) ≤ P(E) + P(F)

Q: (a) Show that for all n € Z, n ≥ 1, if A and B₁, B2, ... are any sets, then (c) n i=1 (b) Show that…

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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: A) Let f: Z→ N as follows: 2n – 1 for n > 0, - f(n) = -2n for n < 0. Show that f is a bijection.…

A: Bijective function

Q: 1. Let S= {2n15 :n€ N} (i) Find the infimum of S and justify your claim. (ii) Find the supremum of S…

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Q: Q3. Write the Disjunctive normal form of Boolean Polynomial p = x (y+z)' + ( x y + z ) x'.

A: Disjunctive Normal form: A Boolean expression is said to be disjunctive normal form (DNF) if it is…

Q: Let Z* be the set of positive integers. We define the binary operation f:Z* x Z* → Q by f(a, b) = .…

A: The binary operation is defined from the cross-product of a set to another set. The binary operation…

Q: Prove that if ø is convex, then a-[$(x + Az) – Þ(x)] is nondecreasing for å > 0.

A: As  φ is convex we conclude that its second derivative :  φ'' is always positive ∴  φ''(x) &gt; 0…

Q: Let P(x) be a property about some object x of type X. If we want to disprove the claim that "P(x) is…

A: Given that : Let P(x) be a property about some object x of type X. And the claim is " P(x) is true…

Q: 11) Let H < K < G. Prove that |G:H|= |G:K| · |K:H| (Do not assume G is finite).

A: Given : H≤K≤G We have to prove that G : H=G : K. K : H

Q: Proof analysis. The following is a proof to a theorem. Read the proof and state what the theorem is…

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Q: Z Prove that the residue of functions and 2³ - 7² +1 Z3 (z-a) (z-b) at infinity are -1 and 1…

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Q: Prove that a set T1 is denumerable if and only if there is a bijection from T1 onto a denumerable…

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Q: Show that there exists a set N C (0, 1) that is not Lebesgue measurable.

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Q: 1. Show that there is no monic polynomial p(2) = 2" +an-12-¹+...+ao such that [p(2)| < 1 for all…

A: To show that there is no monic polynomial p(z)=zn+an−1zn−1+…+a0 such that |p(z)|&lt;1 for all z with…

Q: A field F is said to be formally real if -1 can not be expressed asa su

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Q: Let A = (a, a, as) and B = {b₁,b₂, b3} be bases for V, and let P = {[b₁]A, [b₂]A, [b3]A} (are)…

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Q: Q18 (6) Let a,b, c be three R. elements. and (ab) = 1.g abc then Show that a/c- (b) Let A an d B be…

A: PART A Given that , If a, b, c be the statements of a Euclidean ring R and gcda,b=1 . If a|bc,  Then…

Q: Prove that S = {p(x) = R[x] | p(2) = 0 and p(3) = 0} is a subring of R[x].

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Q: Prove that if x is not free in the formula A, then ` A ≡ (∃x)A

A: The formula is -------(1). To Prove: x is not free in the formula Proof: Let us assume that A(x) :…

Q: Using the special subsets of ordered fields and their definitions,  Prove that: a. If 0 ≠ p ∈  Q…

A: We will prove this using contradiction

Q: True or false (PROVE YOUR ANSWER) :-If A is a set of real numbers that is bounded above and sup A∉…

A: The statement is True.   Proof: Given that A⊆ℝ is bounded above and sup A∉A Let b=sup A Then for any…

Q: For any set A and any ≥0, there exists an open set O such that A CO and m* (0) ≤m* (A) + €.

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Q: 12.- Suppose K and F are disjoint in RP where K is compact and F is closed. Prove that there exists…

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Q: Theorem 6.3. Let a be a real algebraic number of degree n C= = c(a) >0 such that

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Q: Prove that exists integers x, y, and z such that 1/x + 1/y + 1/z is also an integer.

A: Given: The expression is 1x+1y+1z where x, y and z are integers. To prove: There exist integers x,…

Q: RX R:p + q = (e) {(p, q) = R X

A: Given that p,q∈ℝ×ℝ:p+q=1

Q: find all positive integers m satisfying the condition Ø(m)=16

A: find all positive integers m satisfying the condition Ø(m)=16

Q: Prove that S = subring of R[x]. {p(x) = R[x] | p(2) = 0 or p(3) = 0} is not

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Q: *).). (a) Show that X is measurable with respect to o-field G if and only if o(X) C G. Show that X…

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Q: Using the theorem below to construct an alternative proof to the idea that ‘being associates’ is…

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Q: Consider the set 6 A = {3 - z € R, z 2 1}. (a) Find the supremum of A or prove that it does not…

A: A=3-6x:x∈R,x≥1We have to find a) Supremum of A(if exists)b)  Infimum of A (if exists)c) Max(A) (if…

Q: n m+n ) (₁ - ₁) + - + ( ) ( ) = (";"). (™+"). r > a. in valid

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