. Let A CQ satisfying the following conditions: (a) A + Ø and A # Q. (b) If p E A and q E Q with q < p then q E A. (c) If p E A then p < r for some r e A Then prove that: (i) A is a bounded above subset of Q but A does not have supremum in A. (ii) If p E A and q ¢ A then p < q. (iii) If r ¢ A and r < s then s ¢ A.
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- For each of the following statements determine whether it is true or false. No justification is needed. (a) Let X be a finite set and AC 2X be an algebra. Then A is a o-algebra. (b) Let (X, A) and (Y, B) be measurable spaces and f : X → Y a measurable function. (i) The set {f-'(A) : A E B} is a o-algebra. (ii) The set {A E B : ƒ-'(A) E A} is a o-algebra.1. True or False? Prove your answer!Suppose A is a countably infinite set of real numbers that is bounded above, and let S = sup A. Then for every E > 0, the interval (S − E, S) contains a number in A and a number not in A.2.28 let xa be a translation of R Then (1) La is a continuous bijection from onto R (ii) The image of an open set under Da is an oper set. of (iii) let & be an open set. The component interval +a are exactly The images of the component intervals of the set & under translation a the goal of this section is to establish the following result.
- If S is a non-empty subset of R which is bounded below, then a real number t is the infimunm of S iff the followving two conditions hold : (i) x2t V xeS. (ii) Given any ɛ> 0, 3 some xe S such that xLet & > 0. The set {y ER: Ix-yl ≤e} is an open & -neighborhood of x E R. True False An upper bound m of a set S is the maximum of Sif mES. True False A real number m is an upper bound of a set SCR if Vs ES, sLet A and B be subsets of R which are bounded above. a. Show that sup(AUB) = max {sup(A), sup(B)}. Let y = sup (A U B), a = sup (A), and ß = sup (B)Use Cauchy-Schwartz inequality to prove the relation in the attachment.Tick true or false in from with the following: 1- Let Z={1,2,3,4,5} then the power set of Z it's contain 16 element. 2- If P and Q are two proposition, then Q → (~P v P) is contradiction. 3- Let F: R → Z is a function, defined by F(x) = |x| – 3 then F is surjective but not injective. 4- The postfix form the expression (x + y)^ 2) + ((x – 4)/3) is x y + 2 ^ x 4 – 3. 5- if A is a sub set, then ĀnĀ = Ø.Example [2.1.21]| Let Ej and E, are subsets of a space X show that whether each one of the following statements true or fals and why? 1) (E, U E2)° = E,° U E,° ESE 2) (E, N E2)° = E,° n E,° (5,0E)S E,UE 3) (E, U E2)° = E,° U E,° 4) (E, N E2)º = E,° n E2° 5) d(E, U E2) = d(E,)U d(E2) 6) d(E, N E2) = d(E,)nd(E,) 7) (E U E2)' = E,' U E,' 8) (Ε η Ε) = E' n E. 9) (E, U E2) = E, U E, 10) (E, N E2) = E, N E, %3DSEE MORE QUESTIONSRecommended textbooks for youCalculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage Learning
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