. 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.

Q: contrapositive and converse of each statement. (a) If all birds fly, and if Tweety is a bird, then…

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Q: 2. Let x be an element of a convex set S. Assume that x1 = x + €ip E S and x2 = x – €2P E S, where p…

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Q: 2. Decide whether each of the following sets are finite, denumerable or uncountable. Explain each of…

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Q: Let M be the set of all measurable sets and m is the Lebesgue measure on R. Show that if A, B E M…

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Q: 1. (a) Consider the set S = {n(-1)" : n e N}. Verify the existence or otherwise of Min(S) and…

A: a) The given set is,S=n-1n:n∈ℕ

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Q: Ex 13.18. Let A₁,..., Am and B₁,..., Bm be subsets of an n-element set such that A¡ Ñ B¡| is odd for…

A: Given: An element set, with subsets defined as  and .For all , the intersection of and  has an odd…

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A: Given: f:A→A a) a∈fa b) fx∈fy To find: Solution

Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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Q: Which of the followings is a base of the set W = {[x 0 y] : x,y E R}:

A: As per given by the question, There are given that, Then, Find the base of the set W. Now, From the…

Q: Exercises for Sets and Functions Q1) How many elements does each of these sets have? b. P(Ø, {Ø),{Ø,…

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

Q: Let {α1,α2,03,...,an} and {b1,62,63, s. Then 1/2 2 Σ= (Σ})(Σ

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Q: Find an expression for sup in terms of inf E, sup E, inf F, sup F and prove it. 2F

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Q: Prove that the Frobenius norm is consistent. That is, for A, B € Rxn, show that ||AB|| ≤ ||A||||B||F

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Q: f : A → B is a function, X is a finite subset of Z, and k is a positive integer For each of the…

A: Given: f : A → B is a function, X is a finite subset of Z, and k is a positiveinteger. We have to…

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Q: Find the null space for A = 1 2 What is null(A)? {(} O A. O B. R? {} O C. span O D. span: 2 2. O E.…

A: We have to find the nulllspace of the given matrix .

Q: Ex 4.) Let Sbe a non empty bounded set in 1. aj Let azo, and let a S:= {as: SES}. Prove that inf…

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Q: Exercise. Consider the set R?. For every r = (x1, x2) E R? and y = (Yı, 42) E R², write P1(x, y) =|…

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Q: 4. Let EC Rd be a measurable set such that |E| > 0, and assume that ƒ : E → [-∞, ∞] is a measurable…

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Q: Suppose that A is a nonempty set of real numbers, and suppose that a ER is an upper bound for A.…

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Q: Prove or disprove: Given any set S, there exists a function f : P(S) → R that satisfies the…

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Q: (#4) Show that a set SCR" is closed if and only if bdry(S) CS.

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Q: 6. Let z E C. Let C be set of points z satisfying Im(z) 0 and Im Prove that C = {ZEC : |Z| = 1}.…

A: Given that  Im1+z+z21-z+z2=0 We know that if Imz=0 then Im1z=0 Therefore  Im1-z+z21+z+z2=0 Finding…

Q: Determine whether the following relations are reflexive, symmetric, antisymmetric and/or transitive.…

A: To determine: whether the given relations are reflexive, symmetric, antisymmetric, and/or…

Q: 2.6.4 Let A CR be a non – empty set. Suppose that A is bounded above. Let U = {x E R|x is an upper…

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Q: The statement form (ac) ⇒ (bc) is equivalent to O [(~avc) ^ (av~c)] ^ [(~b v c) ^ (b v~c)] ~[(~a v…

A: Since you have posted a multiple question according to guildlines I will solve first question for…

Q: 2.12 let A and B bounded sets For which there is an ε >o such that la-blze for all ac A, be B .…

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Q: Let A, B be bounded subsets of R. Let A+B = {r+y:1 € A, y E B} and AB = {ry : 1 € A, y € B}. Which…

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Q: B. Let S and T be sets of real numbers, and define S. T to be {x.y:x e Sandy € T}. Show that if S…

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Q: Show that: for any z, w in C. ||2|-|w||≤|zw|≤|z| + |w|

A: we have to show that  ≤ ≤ for any z and w in .

Q: 3.7 Consider a fuzzy relation R CAXA where A = {a, b, c}. R a b e 1 1 1 1 1 1 d 1 1 1 a) Determine…

A: Solution :-

Q: Proposition 2.3. Let X,Y be (topological) sets containing E and F, re- spectively. Moreover, let &…

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Q: P(0) = {0} P(A) \ {0} P(A) ACB iff P(A) CP(B).

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Q: Prop 7.7) Let f [a, b] R then, 1) L(f, P) ≤U(f, P) 2) Let P₁ and P2 be partitions of [a, b], then…

A: “Since you have posted multiple questions, we will provide the solution only to the first question…