3. X = {m € N|1 ≤ m ≤ 20}. Y is the set of all nonempty subsets of X. An action of S20 on Y is given by o.U = {o(m) € X/m € U} for any σ € S20 and any U € Y. Determine the number of orbits and the number of fixed points for this action.

Q: Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7. (i) Prove that…

A: Given that X,Y,Z are subsets of {1,2,3,...,10} |X|=|Y|=|Z|=7. X∪Y≤10 i)X∩Y≥4 X∪Y=X+Y-X∩YX∩Y=X+Y-X∪Y…

Q: ercise 6.67. Let E be a subset of R having at least two points. Pro- owing are equivalent:

A:

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

A: given that (1) (a)                                    S=n(-1)n:n∈N=2k(-1)2k:n=2k,…

Q: Suppose that S1, S2, ..., Sn are sets in R'; and that S=S1∪S2∪ ...∪Sn Define Bi= sup Si and bi=inf…

A:

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…

A:

Q: 1. Let Ω = (0, ∞). Let A = (3, 8] and B = (5, 10]. Find the smallest collection of subsets of Ω that…

A: Definition: Let Ω be a non-empty set. Let Σ be a collection of subsets of Ω. The collection Σ is…

Q: 7. Let x E R and A be a nonempty subset of R. If x is not a limit point of A then x is an isolated…

A:

Q: Find an example of a closed convex set S in R2 such that its profile P is nonempty but convP ≠ S.

A:

Q: Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.

A:

Q: Consider the set T = {..., −34, −17, 0, 17, 34, ... } with the usual definition of less than (a0).…

A: Completeness property of real numbers:  Every non-empty set of real numbers that has an upper bound…

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.

A:

Q: 1.2 Now consider the following sets: X {(x, y) : x² + y² < 4} (5) %3D Y = {(x, y) : -x +y < -2} (6)…

A: Given: X = x, y : x2 + y2 &lt; 4Y = x, y : -x + y &lt; -2

Q: Exercise 6.41. Let E be a nonempty set in Rn. Then the distance of a point x € R¹ to E is defined by…

A:

Q: 1. Let {v₁,..., Vn} be a linearly dependent set. Prove that there exists some j with 1 ≤ j<n such…

A: Suppose that    be a  linearly  dependentset.Then there exists some   such that where  are  scalars…

Q: Q.No.3 a) Prove that if A and B are sets, then (An B)U (ANB') = A

A:

Q: (3) S° is the largest open set in M that contained in S.

A:

Q: Let U {1,2,..., 100}, and let A consist of 15 unknown numbers taken ran- %3D domly from U. Let B…

A:

Q: Given the following fuzzy relations 0.2 0.4 0.6 R₁: 0.1 0.9 0.0 0 0.4 0.8 Compute the following 1.…

A:

Q: 2. For each of the following sets, determine whether each of these statements is true or false.…

A:

Q: 4. Let Y + 0 and let A be a collection of subsets of Y satisfying the following two properties: (a)…

A:

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

A:

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: Let An denote the number of lattice paths from (0,0) to (n, n) that do not pass above the diagonal y…

A: As per the norms, we will be answering the first question. If you need an answer to others, then…

Q: a. Write o using the two row form. b. Write o using cycle notation. c. Write the function o¬1 in the…

A:

Q: second and thrid answer are incorrect

A:

Q: 3. Let X be the set of positive integers. For each n E X, let Sn = {k € X: k2n}. Show that T = {S:…

A:

Q: = (4) Suppose that A is nonempty set that is bounded from above and let B Prove that B is bounded…

A:

Q: 4.4.4 List the relation on N defined bv h(x, y) e N x N : x > y } 9 Scanned with CS CamScanner as a…

A: The relation given is already a subset of the Cartesian product N x N. It can be written as follows.

Q: 2 49) Given nonempty subsets of R², say Y....., let Y = nin enner-xer Fix P E R². For a nonempty…

A: Given: Non empty subset To find: Solution

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 .…

A: Consider A and B are bounded sets.

Q: Let V C(R ) be a nonempty open set satisfying a – b EV whenever a, b e V. Prove that V = R. %D

A:

Q: Let S = {1,2,3, 4}. For each of the following relations on S give its transitive closure. a) {(1,1),…

A: Concept: In mathematics, the transitive closure of a binary relation R on set X is the smallest…

Q: 4. Let X be an ordered set, z E X, and SC X. Prove that x = sup S, if and only if Vy EX (yy).

A:

Q: 1. Find all sets A such that a. {1} C A and A C {1, 2, 4} b. {1, 2} CA and A C {1,2, 4, 8}

A: see below the answer

Q: (c) Let X, Y, Z, and T be arbitrary sets as per the diagram below, with U being the universal set.…

A:

Q: If geRla,b] and if h(w)= gcu) encept %D finite number f points for [a, b],'then and in of prove that…

A: First we have to divide the set into finite subsets of x intervals required.

Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

A:

Q: (c) Determine whether the following sets M C R are invariant, positively invariant, negatively…

A: The complete solutions are given below

Q: (3.2) Indicate which of the following sets are compact in X. (a) C = {4,n € N}, X = R (b) C = Qn (0,…

A:

Q: Let A = {x + iy : x 0}. Determine the set of all z for which A. (Hint: this is {z : Re() 0}. Find…

A: Set of Z found.