Consider the set A={a,b,c,d,e,f}. Define a partial ordering on A by setting x<=y if (x,y) belongs to R={(a,a)(b,b)(c,c)(d,d)(e,e)(f,f)(a,d)(a,f)(b,c)(b,d)(e,b)(e,c)(e,d)(f,d)} a)Find all x belongs to A that satisfy x

Q: We say that a space (X, τ) is not connected if there exist two nonempty openings A, B such that A ∩…

A: Open Set: A X, τ be a topological space. Then the sets in τ are called the open sets of the…

Q: Let X = {1,2, 3, 4} and let T = {, X, {1}, {3},{1,3}} be a topology on X, then {2, 4} is a. open in…

A:

Q: Define spanning set. Determ u2 = (-1,1,2) and Uz = (3, %3D

A: Define spanning set:

Q: Let X = {a, b, c, d, e, f, g} and B = {{a,c, e, f}, {b, c, d, f,g}, {c}, {f}}. Show that B is a base…

A:

Q: A ° = A – JA

A:

Q: Exercise 2.46. Show that: (1) The topology generated by a subbasis is a topology. (2) The sets in…

A: 2.46

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

A:

Q: Prove that every well order is a linear order,

A: A set  is said to be well-ordered if it has a total order and has the property that every non-empty…

Q: Let R be a partial ordering on a set A, and Ba subset of A. If B has a greatest lower bound (an…

A:

Q: Let X={a, b, c, d, e} and let T={X, non zero, {a}, {a, d}, {a, e}, then (X, T) is neither Hausdorff…

A:

Q: Find the transitive closure of R={(0,1),(0,2),(1,1), (1,3),(2,2),(3,0)}

A: Given relation is                              R=(0, 1), (0, 2), (1, 1), (1, 3), (2, 2), (3, 0) we…

Q: A player has two continuous choice variables: x € [0, 1], and y E [0, 1] Illustrate the player's…

A: The solution is given below in the next step:

Q: 2. Is there a set A that satisfies A = {A}? If yes, exhibit one %3D such. If not, Why not exactly?

A: No, A≠{A} for any set A. Let  A=1,2,3 then number of elements in A are 3. Now,  A=1,2,3 here the…

Q: If X is a normed space then the open ball and closed ball is a Example: convex set

A:

Q: Each nonempty can be represented as the convex hull of its extreme points. --------

A: Each of the nonempty compact convex set can be represented as the convex hull of its extreme point.

Q: Let X = {a, b, c, d,e} and let T = {X, ø, {a}, {a,d}, {a,e}}, then * %3D (X, T) is not Hausdorff but…

A:

Q: = {a, b, c, d}, t = {Ø, X, {b, c}}. Then in the topological space (X, t), {a} = %3D %3D %3D {a, b}…

A:

Q: be a filter in a space X and v be the associated net

A:

Q: 3. Suppose that {v,,v,,v,} is a linearly independent subset of R". Show that the set {v,,v, +v3,v; +…

A: The set {v1,v2,......vn} is linearly independent if for the combination, av1+bv2+......nvn=0 for…

Q: Is the set X is linearly dependent or independent X = {(1,0,1,0), (0,3,0,1), (1,1,2,2), (3,4,1, −2)}

A:

Q: Prove that the convex hull of a bounded set is bounded.

A:

Q: Let X = {1,2, 3, 4} and let T {ø, X, {1}, {3},{1,3}} be a topology on X, then {2, 4} is a. open in…

A: Our objective is to choose the correct answer

Q: Provide an example of a topological space that is connected but not path-connected,why?

A: We use definition of connected and path connected sets. We use contradiction method.

Q: A(n)............ is the collection of all points in the plane the sum of whose distances from two…

A: It is given that,   A(n) ............ is the collection of all points in the plane the sum of…

Q: The set S=[1,3) is not compact. Which open cover of S contains a finite subcover?

A: This is the question of Topology.

Q: Prove or disprove: If R and S are partial orderings on A, then R ∩ S is also a partial ordering on…

A: Partial order relation of intersection of two partial order set

Q: Determine if the set is linearly independent. S = {(4,0,1), (0, -3,2), (5,10,0)}

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: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

A:

Q: Suppose that {u, v} is a linearly independent set. Show that the set {u – v, u + v} is also linearly…

A:

Q: Show that if a topological space has a finite number of points each of which is closed then it has…

A: Topology question.

Q: Suppose the collection = {B₁ (x): xe (0,1)}. Is B an open cover of the interval [0, 1) CR? Give an…

A: Given: The collection B=B12x : x∈[0, 1). To determine: Whether B is an open covering of the interval…

Q: Decide whether the following relations is a partial order or not. (a, b) | a Í b, a and b are…

A: We have to decide whether given relation is partial order or not?

Q: {(a, b) | a < b, 2<a<5, 2<bs5}

A:

Q: Let R = {(0,1), (0,2), (1,1), (1,3), (2,2), (3,0)} Find R', the transitive closure of R

A:

Q: Exercise 4.5. Let R be a irreflexive and transitive relation on the set S. Then R is a strict linear…

A: A relation is a strict order on a set   if it is 1. Irreflexive:  (a,a) ∉ R for any . 2.…

Q: 2. Let a,, az,.., a, be numbers. Show that there is a single number r such that foxo: (-a,) + (z –…

A: Solving

Q: Suppose Ak,., A2 , A1 and B are points on R" Prove that if {A1 , A2, .. Ak} is an independent linear…

A: proof :- A1,A2,...Ak are linear independent set ⇒c1A1+c2A2,...ckAk=0  where c1=c2=...=ck=0…

Q: 4. Let R be a relation on the set {1,2, 3, 4} such that R = {(1, 1), (1, 3), (2, 2), (2, 3), (3,2),…

A:

Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

Q: Let X = {a, b, c}. Is the collection {X, Ø, {a}, {a, b}, {c}} a topology on X?

A: Here, the given set is X=a,b,c and collection is X,∅,a,a,b,c.

Q: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

A: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

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: Let, s be the circle given by, s=x,y∈ℝ2 :x2+y2=2  Then s is a closed subset of ℝ2. Let us consider a…

Q: Let F(x, y, z) = (z, x, y), ar (0,0,0), (0, 1, 1), vertices are ar

A: Introduction: Stroke's theorem is simply a higher-dimensional version of Green's theorem. The line…

Q: For any partial order there may be a number of linear orders that are consistent. For example, for…

A: This is a question from Relation.

Q: If R = {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2)}, what is the symmetric closure of R? %3D О (1, 2),…

A:

Q: Suppose {u,..., 4,, W1, ..., W,} is a linearly independent subset of V. Show that span (u;)…

A:

Q: How many different partial orderings can we define on the set A = (x, y}? How many different total…

A: 1) The partial ordering of a two-element set must contain the following conditions: a) Reflexive…

Q: Let (A, B, C, D) be an ordered quadruple of distinct collinear points in II. Let G = S4((A,B,C, D))…

A: Given that (A,B,C,D) is an ordered quadruple of distinct collinear points.   Also given a set…