2) Let a,bE Q such that a<6. Prove that ax

Q: Prove that if alb and alc then a (bx + cy) for all integers x, y.

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Q: Suppose A and B are bounded non-empty sets of real numbers. Prove sup ( (AUB) max & Sup A sup Bee or…

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Q: Prove: | a | - | b | &lt;= | a + b |

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Q: 3) Prove that ae*+be-x a) b b > 0 ex+e~x

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Q: 2. Show that for any real number c, there is exactly one integer in the interval (c,c + 1].

A: As per guidelines, I can do only first question.

Q: |x²| = p^ n

A: Given: X=pWe need to show that Xn=pn

Q: Let A be a subset of real numbers bounded from below. Then, the set-2A = {-2x: x E A} is bounded…

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Q: Pzif akb then [a,b]-(- All of a,b are in real numbers

A: The given statements are: p: a&lt;b q: a, b=(-∞, b] \ -∞, a Prove: If p, then q. If a&lt;b, then  a,…

Q: Let x1, x2,...,n be real numbers. Prove that |X1 + X2 + ··· + Xn] ≤ |X1| + |X2| + · · · + |xn|·

A: The objective of this question is to prove the triangle inequality for a sequence of real numbers.…

Q: Suppose x and y are integers. Show that If x y, then x | 3y - y + 5y. 1) What is your assumption? 2)…

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Q: 4) Let a < b be real numbers. Prove that the half open/half closed interval (a, b) is both an Fo and…

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Q: XA and x & f(x)}. Show Cim ƒ.) 38. Let a, b, c, and d be any real numbers such that a < b and c < d.…

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Q: 7. (12 points) Use a proof by contradiction to prove: For all sets A, B, and C (A\C)n(B\C)n(A\B) = 0

A: Step 1: proof  let (A/C)∩(B/C)∩(A/B)=Φ this means ∃x∈(A/C)∩(B/C)∩(A/B) Step 2:⇒x∈A/C,x∈B/C,x∈A/B…

Q: x + y)(z + w) = xz + xw + yz + yw A.) Explain in your own words what this problem is asking. B.)…

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Q: mutually disjoint? Why or why not?

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Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

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Q: 2. For all real numbers a, b and c, prove that if a <b <c then a+b+c a < 3 < C.

A: given that a, b, c are real numbers we have to prove that if      a&lt;b&lt;c    then…

Q: Let a, b, c. and d be any real numbers such that a < b and c < d. Prove that [a, b] is equivalent to…

A: Answer

Q: Lise a direct proof to show "fx's odd and y is even, anen x+1s odd."

A: The given statement is: "If x is odd and y is even, then x+y is odd" Result: Any odd number x can be…

Q: Prove that ord,(a + b) > min(ord,a, ord,b) with equality holds if ord,a ± ord,b.

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Q: Prove that for any set ACR and any real number a we have m* (A +a)= m* (A).

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Q: Suppose that x is a real number and that UCR. Prove that the following two conditions are…

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Q: Let U= {x|x EN and x<7) A = {x|x is an even natural number and x<7} B=(x|x is an odd natural number…

A: Given problem:- Let   U={x|x ∈ N and x &lt; 7} A={x|x| is an even natural number and x &lt; 7}…

Q: 3. (10 points) a) Use an element proof to prove the following: for all sets A and B, A - (ANB) = A –…

A: The detailed solution is as follows below:

Q: 3) Let f(x) = 1 + 3x on [-8, 2]. Find the number c that satisfies the mean value theorem. a) O C= b)…

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Q: Let a, b e C. (i) Prove that if |a| <1 and |6| < 1, then a - b < 1. 1- āb

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Q: Which of the following defines the domain of |4x-8<12? A) -8<x<-4 B)-4 <x < 8 C) -1<x< 5 D) 1<x<5

A: We can solve the inequality in order to get the solution.

Q: Give an example of a closed subset S of R2 such that conv S is not closed.

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Q: 5. Let x and y be irrational numbers such that x-y is also irrational. Define sets A and B by A = {x…

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Q: Prove: For c &gt;= 0 , | a | &lt;= c if and only if -c &lt;= a &lt;= c

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Q: 1. Show that if P(A) > 0, then P(AB|A) > P(AB|AUB)

A: Show that if P(A) &gt; 0 thenP(AB|A) &gt; P(AB|AUB)

Q: Let Xipnnig Xu~N(Po²). Find Consistent estimator of M using a chebychev's inequality.

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Q: Let A=(aij)nxn be symmetric and positive definite. Show that laul< (anan), , lis = 1,2,..n 9

A: Given: Let us consider the given, A=aijn×n Let's prove that aij&lt;aiiajj12,i,j=1,2,...,n.

Q: 2 if f) = k x?,-1<x<1,then k = true false

A:

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”.