Q4. Let X = {m, n, p, q},T = {ó, X, {p}, {m,n}, {m,n, p}} and Y = {a, b, c, d}, F = {ó, X, {d}, {a, b}, {a, b, d}} Given the mapping f : (X,7) (Y, F) by | f(m) = a, f(n) = b, f(p) = d, f(q) = c. Is fa bijection? Is f a continuous mapping? Is f an open mapping? Is f a homeomorphism ? What you can say about the topological spaces (X, 7) and (Y, F). ( Show the details of your answers).

Q: Let X = (a, b, c, d) and TX = (Ø, X, (b), {b.c), {a,b,d}) be a topology 3, 4) and TY = (0, Y, (3),…

A:

Q: Let X={a,b,c} and Y={1,2,3} and t={Y,¤,{1,2}}. The function f={(a,1),(b,3),(c,2)}. Then the topology…

A:

Q: Let X:= (a, b.c), n=(1,2,3.4), Ty:= (0,Y, {1,3), [3,4}, {3), [1,3,4}}. Let fx-Y. f:= {(a,3). (b, 3).…

A:

Q: given two sets X={-1,0,1} and Y={u,v,x,y,z}. There are 125 mappings from X to true or false

A:

Q: b) Is there a linear map T: V → W, where dim V = 3, dim W = 4, rank T = 2, and dim ker T = 1?

A:

Q: 4. Suppose that T : R* → R² is a linear map defined by T (x1, x2, x3, x4) = (2x1 x2 + x3, -2x1 – x2…

A:

Q: 3: Show that if T : R³ →R is a non-zero linear map, then it must be given by T = ax + by + cz for…

A:

Q: 15. If t={X,Ø,{1,4},{4,3},{1,4,a},{b}} is a topology on X-{1,2,3,4}, then: * a=4, b=2 O a=3, b=1 O…

A:

Q: 10. Let (X, Tx) and (Y, Ty) be two topological spaces, such that X = {a, b,c, d}, Tx = {0, X, {a},…

A: Definition: A function f is continuous from X to Y if inverse image of each open set in Y is open in…

Q: If (x, y) → (x +3, y- 7)maps TUVWto T'U'V'W', what is the algebraic description that will map…

A: Consider the given: Mapping TUVW to T'U'V'W':x,y→x+3,y-7 Find the algebraic description that will…

Q: Given a linear map L:R-R such that L(1,0,0)=(3,-1),L(0,1,0)=(2,1),L(0,0,1)=(3,0) , then find…

A:

Q: Given the map f: R3 R4 defined by f(x,y,z) = (0,0,x,0) then rank of f is: O A. 3- O B. 2- OC.- OD.…

A:

Q: Show that the map T : R3 -→R? given by T (x1, X2, X3) = (x1 + x2, X3) is an open map.

A:

Q: 4) B. Let A = {u, v, w}, B = {x, y, z}. Determine whether the relation R = {(u, x), (v, z), (w, y)}…

A:

Q: Let (X, Ox) and (Y, Oy) be affine varieties over K. Recall that we have constructed the affine…

A: Given: Let (X, OX) and Y, OY be affine varieties over K. Now, we have constructed the affine variety…

Q: 4. Consider two maps S,T: R" R". The composition So T is a linear map if and only if both S and T…

A: 4. Consider two maps S,T:IRn→IRnThe composition SoT is a linear map ifboth S and T are linear maps

Q: a) Is there exists a linear map T: R² → R² for which T(7,7) = (4,-5) and T(2,2) = (8,-2)? Justify…

A: As per our guidelines, we are supposed to solve only first subpart. Kindly repost other subpart as…

Q: Let X = {a, b, c, d} and TX = {ø, X, {b}, {b,c}, {a,b,d}} be a topology on X. Let Y = {1, 2, 3, 4}…

A:

Q: Need help with Discrete math, in specific Relations

A:

Q: For each n E N, we write S" := {(x1, x2, ..., In+1) E R"+1 : E? = 1}. (1) Let f : S2 → Rª be the…

A: Given: Sn:=x1,x2,...,xn+1∈ℝn+1:∑i=1n+1xi2=1 for each n∈ℕ f:S2→ℝ4 is the function defined by…

Q: Let X = {a, b, c, d} and TX = {Ø, X, {b}, {b,c), {a,b,d}} be a topology on X. Let Y = {1, 2, 3, 4}…

A: A map is continuous if and only if inverse image of every open set is open

Q: Consider X = [1,2]? with the relation (x1, y1)~(x2,y2) if x1y2 = X2y1. Also define g: X [1,2] by the…

A:

Q: 9. Let X = {r1,12, 23, T4}, Y = {yı,y2, Y3}, and Z = {zı, z2}, Let R be the relation from X to Z…

A: Given: 9) X=x1, x2, x3, x4, Y=y1, y2, y3 and Z=z1, z2, z3. R=x1, z1, x2, z2, x3, z1, x3, z2 and…

Q: 15-18: Suppose we define a relation T from A = {-3, 2} to B = {-2, 0, 1} as follows: For all (x, y)…

A:

Q: 2. Compute P, and Py as a function of x, y, P.

A:

Q: 5. If (x, y) → (x+3,y-7)maps TUVWto TU'v'W', what is the algebraic description that will map…

A: As per rules we will solve only first question: 5. Given that (x,y)→(x+3,y-7) maps TUVW to T'U'V'W'…

Q: Let X = {a, b, c, d} and TX = {Ø, X, {b}, {b,c}, {a,b,d}} be a topology on X. Let Y = {1, 2, 3, 4}…

A:

Q: then Given the map f: R³→R4 defined by f(x,y,z) = (0,0,x,0) rank of f is: А. В. 3. Е. 2. 4-

A:

Q: please solve this problem

A: Consider the provided question,

Q: The image of the line Im(z) = -2 under the mapping f(z)= i(z) is

A: Solution:

Q: Let X = {a, b, c, d} and TX = {ø, X, {b}, %3D {b,c}, {a,b,d}} be a topology on X. Let Y = {1, 2, 3,…

A: A function is continuous if and only if inverse image of every open set is open

Q: 3. Define a relation S from R to R as follows: For all (x, y) e R xR, (x, y) eS means that x2 y. a.…

A: Given x,y∈S  Mean x≥y    x,y∈R×R

Q: Let G = {-2, 0, 2} and H = {4, 6, 8} and define a relation V from G to H as follows. For every (x,…

A: Given,            G=-2, 0, 2  and   H=4, 6, 8and  define a relation v from G to H as follows.For…

Q: 11.1. Find images of the following sets under the mapping w = e² : (a). the vertical line segment x…

A: As per our guidelines we are supposed to solve only first three subparts of the problem. For the…

Q: If n<m then every linear map T: IRn -> IRm is one-to-one. Is this statement true or false?

A: The given statement is

Q: Let A = {u, v, w}, B = {x, y, z}. Determine whether the relation R = {(u, x), (v, z), (w, y)} from A…

A: In R from A to B is an injection function as well as a surjection function.

Q: Let X= (a, b, c, d) and TX = {Ø, X, {b}, {b,c}, {a,b,d}} be a topology on X. Let Y= 3, 4) and TY=…

A: A function is continuous if and only if inverse image of every open set is open

Q: Let X = {a, b, c, d} and TX = {ø, X, {b}, {b,c}, {a,b,d}} be a topology on X. Let Y = {1, 2, 3, 4}…

A:

Q: 8. Let A (2, 4) and B (1,3, 5) and define relations U, V, and W from A to B as follows: For all (x,…

A: Given that: A={2,4}B={1,3,5}

Q: Consider X = [1,2]? with the relation (x1, y1)~(x2, Y2) if x1Y2 = x2y1. Also define 3x g: X → [1,2]…

A:

Q: Which of the following maps is not linear? O A. L-R2-R3 defined by L(x,y) = (x+y,y.x)- OB. K:R R3…

A:

Q: 10. Let (X, Tx) and (Y, Tr) be two topological spaces, such that X = {a, b,c, d}, Tx = {ó, X, {a},…

A:

Q: on 6 TOTO Mark Let M = R, and f: R→Y be a continuous map. Let A, BCY and K = A UB. (a) Give f-¹(K),…

A: here we can use the properties of continuous functions.

Q: Let A = {2, 4} and B = {1, 3, 5} and define rela- tions U, V, and W from A to B as follows: For…

A: (a) First obtain the 3 relations as follows.

Q: Let f: X - Y and g : Y - X be maps between sets. Suppose that g•f is equal to the identity map idy :…

A:

Q: 4. Consider two maps S, T: R" R". The composition So T is a linear map if and only if both S and T…

A:

Q: Let T : R? →→ R² and S : R² → R² be a linear maps such that r() - (1) (G) = (;) TO and S(( 2 1 Then…

A: Given: T: R2→R2 ,  S: R2→R2 be linear maps such that T10=11 & S11=12 To determine: S∘T10

Q: (2) Suppose UC R" and V C R" are open subsets, and we have C maps U V 4U such gof is the identity of…

A:

Q: Let X = {a, b, c, d} and TX = {ø, X, {b}, {b,c}, {a,b,d}} be a topology on X. Let Y = {1, 2, 3, 4}…

A:

Q: Let (X,Ox) and (Y,Oy) be affine varieties over K. Recall that we have constructed the affine variety…

A: Let X, OX and Y, OY be affine varieties over K . In general the tensor product of two affine…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.