Let P* be a refinement of a partition, P. Show U(P*, f, a) < U(P, f, a)

Q: For the subset "The set of points on the line given by x+2y=0" verify whether it is a sub space of…

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Q: Find the limits in Exercise

A: Given,          limh→0-h2+6h+13-13h

Q: 1. Let A = {a, b, c}, B = {1,2,3}, C = {x, y, z, w}, D = {4,5, 6}. Let the functions f : A → B, g :…

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Q: The relation R defined on a set A is antisymmetric if (a, b) ER = (b, a) E R for (A) Every (a, b) ER…

A: A relation R is said to be antisymmetric if (a,b)∈R and (b,a)∈R implies a=b

Q: X, Y and Z be subsets of the set . Verify the following propert XU(YUZ) = (XUY)UZ, XN(YnZ) = (XnY)…

A: X, Y, Z be subset of a universal set Ω to verify…

Q: For the partial ordering p = {(x, y): x|y, x ∈ S, y ∈ S} where S = {2, 3, 4, 5, 15, 32, 60} List the…

A: Consider the provided question, A relation R on a set S is called a partial ordering if it is…

Q: True or false? For all sets A and B,  (A∪B)'=A'∪B' This can be determined by constructing a…

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Q: Define relations S and R on the set A = {x, y, z, w} as follows S = {(y, x), (x, w), (w, z), (x,…

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Q: et Fn be the number of partitions of [n] that do not contain a singleton block. Find a simple…

A: Given: To find the simple relation between the numbers Fn and the bell numbers.

Q: 4. Let R be the relation in the positive integers

A: 4. Given, R be the relation in the  positive integers ℕ defined by                        R=x, y: x,…

Q: b) A function g: Z → Zis defined by the rule g(m, n) = m + n Determine whether it is onto or not..

A: We will answer the first question as we don't answer multiple questions at a time. Please resubmit…

Q: How would you show g is a contraction map?

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Q: Let A={ 1, 2, 3, 4, 5} R={ (1,1), (1,2), (2,1), (2,2), (3,3), (3,4), (4,3), (4,4), (5,5) } Write the…

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Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

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Q: Let W = (0, 1, 3}, X = {1, 3, 5), Y = (3, 4} and Z = {(1, 2), (3, 4), (1, 3),(2, 4). Compute (X x Y)…

A: The given problem is to find the Cartesian product of given sets and verify the Cartesian product…

Q: Let T={X, {a}, {a,b).{a,c,d}, {a,b,c,d}, {a,b,c}} be a topology on X=(a,b,c,d,e). (ii) Find N.…

A: Given : (X, T) be a topological space,  where  X = {a, b, c, d, e} and T = {X, ϕ, {a}, {a, b}, {a,…

Q: Prove that if f: R→ R is continuous, then the image of a bounded set over f is bounded.

A: As per the question we have to prove that if any function f : ℝ → ℝ is continuous, then the image of…

Q: Let X = {o, p, s, w}, Y = {a,b, e, g, k}, and Z = {1, 2, 3, 5, 7, 9}. Let ƒ : X → Y where f = {(o,…

A: X=o,p,s,wY=a,b,e,g,kZ=1,2,3,5,7,9f:X→Yf=o,a,p,g,s,e,w,bg:Y→Zg=g,3,e,2,b,9,k,7,a,1

Q: Assume that E, F, G₁, G₂, G., G (wherein i = 1 ton) such that P(EG) is less than or equal to P(FIG)…

A: To show that P(E) ≤ P(F), we can use the law of total probability. We know that the G's form a…

Q: If X = {a, b, c} , Tx = {0, X, {a},{b, c}}, Y = {x, y, z} and ty = {Ø, Y, {y}} then f = {(a, z), (b,…

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Q: Define a critical number of f

A: Critical number of a function is point where graph of function changes its direction. Critical…

Q: If S, S, are two subsets of an inner product space V(F) then show that

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Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

A: Inverse image of closed set is closed. Whenever f is continous from that we solved this question.

Q: Prove that if f is a constant function on [a, b] then f = S(f,P) for any tagged partition P of [a,…

A: This is a question of R integration.

Q: Let f:X → Y is a function and let {Ba: a e A} be an indexed family of Subset of Y. Show that a)…

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

Q: 6. For A, B, subsets of R, define A+B= {a+b : a € A,b € B} and A·B = {ab : a € A, b E B} (a) For A =…

A: Answer: As per guidelines we can do only 3 subparts of question and the solution of this is given…

Q: Let A={1, 2, 3, 4, 5} and B={a, e, i, o, u}. If ƒ: B → A is a function defined by f ={(a, 4), (u,…

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Q: Let X = {a, b, 1, 2, 3} and T = {X, p, {a}, {1,2}, {a,1,2}}. Let B = {b,3} * O B is dense in (X,T) O…

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Q: Let E = [-5,20), be the universal set and Y =[-2,4], and Z = (0,10]be the subsets of %3D %3D Find:…

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Q: If (X,7,) and (X,7,) T. - spaces, then (X,7,nt) is also a T,- space. GOOD LUCK

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Q: Let S = {0, 1, 2, 4, 6}. Test the following relation whether it has equivalence relation on S. R =…

A: Given relation R = {(0, 0), (1, 1), (2, 2), (4, 4), (6, 6), (0, 1), (0, 2), (1, 2), (4, 6)} on  S =…

Q: Consider the subset R = R × R − {(x, x)|x ∈ R}. What familiar relation on R is this? Explain.

A: Concept used: Subset relation Subset relations are used to define partial orderings on sets. A set's…

Q: Use equivalences to construct a prenex disjunctive normal form for the following wff. Show your…

A: Given: ∀x∃yp(x,y)→∃y∀p(x,y)