F, there is a disconnection A, B of F such that x E A and y E B. 8.X. If C1, C2 are connected subsets of R, then the product Cı X C2 is a con- nected subset of R?.

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: (B): Let a:M → N, B: N →M are R-homomorphism such that a oB = In then M = Ker(a) + Im(B) and Ker(a)…

A: The problem is related with R homomorphisms. Given: α :M → N , β :N →M are R homomorphism, such that…

Q: 0:C' →C',o(x) = x*ANYWAY. O SOW THAT THERE IS A HOmomoRPHISm AND KER(Ơ) DETERMINE

A: Group Homomorphism: A Homomorphism ϕ from a group G, ∘ to a group G¯, ⊛ is a mapping from Ginto G…

Q: 29. Let N be with the co-finite topology If A={1,3,5.), then Bd(A)=

A:

Q: 6. Sketch the set X = {(x, y) E R? : y<x²} on R2. Shade in the set X. %3D

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Q: B if there exists o E Sn so that oao- = B. Prove that For a, B E Sn, let a ~ equivalence relation…

A:

Q: It follows by definitic ecause F is onto, there exists an x in X, such that F(x) = y image of a set…

A: Everything You have done was correct except the second last step . So, I m doing from there . Since…

Q: Let ficxT)→(メッテ)and g: メっテ)→(Kっズ) e are homomorphism then gof 1s homo.

A:

Q: Suppose that o is a homomorphism from Z3, to Z30 and Ker ø = {0, 10, 20}. If (23) = 9, determine all…

A: Given: ɸ is a homomorphism from Z30 to Z30. Ker ɸ = {0, 10, 20}.

Q: know there exists at least one c in the open values of c that work.

A: We have to find the following conditions according to the given information

Q: 7. Let X = {a,b,c}. Show that Bß ={{a,b},{b,c}}cannot be a base for any topology on Х.

A:

Q: Use a membership table to prove the following relation: * (An B) U (A n Bº) = A

A: We have to use membership table.

Q: Z6 is isomorphic to U(7) by giving an isomorphism φ from Z6 to U(7).

A:

Q: Let A be a subset of R and let c e R. Explain what it means to say that c is a cluster point of A.…

A:

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: Q) Make up a Kelly table for set G= [1,-1, i,-i} with respect to multiptication and then explain…

A:

Q: Suppose A,S CU, (<) e LOrd(A), and B (a1,b1), (a2, b2) E (A × B): (S). If we define a relation (3)…

A: Suppose x=a,b∈A×B is an upper bound of the two point set. We have show that…

Q: (1) Let X={a, b, c}, Let r-{0, X, {a}, {b}, {b, c}}. Which of the following is an open cover of X:…

A:

Q: Show that the mapping :S3 S3 given by p(x) = xis not a homomorphism.

A: Solution is below.

Q: given two sets X={-1,0,1} and Y={u,v,x,y,z}. There are 3x(3-1)x(3-2)%36 onto mappings from X to Y.…

A:

Q: Let G=U(16),H={1,15},andK={1,9} . Are H and K isomorphic?Are G/H and G/K isomorphic?

A:

Q: 1. Decide if the following statement is True or False: %3D Let A1, A2, ..., Ak E Mnxn(R) be such…

A:

Q: 4. Let r E R" be such that r = (E1,... , En). Show that the norms ...+ En and = V are equivalent so…

A:

Q: Let S be the set of integers. If a, b E S, define aRb if ab &gt;= 0. Is R anequivalence relation on…

A:

Q: 2.3 Let 9: (M2(R), +)→ be defined by e c d a b = a +d. where a,b,c,d €R (a) Prove that e is a…

A: A mapping f:(G,+)→(H,+) is said to be group homomorphism if it satisfies: f(a+b)=f(a)+f(b)…

Q: If X = {a,b, c}, let T1 = {0, X, {a}, {a,b}} and T2 = {0, X, {a}, {b, c}} . Find the smallest…

A:

Q: (b) Let A, B, D, E and F be subsets of R defined as follows: {x € R: x 5}, D={xE R: x 6} and F =…

A:

Q: Let Xbe a nonempty subset of R. Show that if c E R is a cluster point of X, then for any n EN, there…

A:

Q: f: (A, SA)(B,S) be an order isomorphism and, Sa={xEA: x<naf## and Spa) =y eB= y <gf(@)}. 3) Let %3D…

A: Consider the given information: Let f:A,≤A⇒B,≤B be an order isomorphism. Sa=x∈A: x&lt;Aa≠∅ and…

Q: Show the following: a) The translation of any open subset of R is open. (Hint: x+tt is…

A: Let  B is subset of R and which is open ,now take  a translation  T:(x + t)  ↦ t  where  x, t ∈B  T…

Q: Let X = {a, b, c, d, e} and let T ={X, Ø, {a}, {c,d}, {a,c,d}, {b,c,d,e} }, then O XT) is neither…

A: Option 1 is correct.

Q: (x), we - Let (-) be the evaluation inner product on P₂ at -1, 0, and 1. That is, for any p = p(x)…

A:

Q: the only homomorphism f : S, to Z,, is the trivial one, Select one: O True O False

A:

Q: Let A, B, C be sets. Prove that if f: A → B and g: B → C are bijections, the composition gof: A → C…

A:

Q: 11.2. Find images of the following sets under the mapping w = 1/z: (a). {z :0 < |z| < 1, 0< arg z <…

A:

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: Suppose that Φ is a homomorphism from U(40) to U(40) and thatKer Φ = {1, 9, 17, 33}. If Φ(11) = 11,…

A: To Determine:  find all elements of U(40)that map to 11 Given:  Suppose that Φ is a homomorphism…

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: In the semigroup with identity (R,) we define Va E R+, fa: (R, +) → (R, +) by Vx € R, fa (x) = ax.…

A: The given map is fa:ℝ,+→(ℝ,+), defined as fa(x)=ax, ∀x∈ℝ. We have to check the following: (a). Is…

Q: Let 2 z, and Z2 = %3D ) show {2, ,2z} that or thonormal c's an set in O Write the rector as a 2 = -…

A:

Q: Discrete math graph theory

A: Two graphs S and T are isomorphic if there exists a bijection f between the vertex sets of S and T…

Q: Let 9-(a: a€R}, then T is Borel algebra.

A: Clearly the measure of the set is =0 .

Q: Consider the subset G {r• /2: x € Z} of R. Show that G is closed un- der addition.

A: The given subset is: G= x·2 : x∈ℤ

Q: a) Let * be the operation on Z defined by a*b= a + b - 3ab for all a,b e Z. Justify the following…

A:

Q: 2. Write a preference E on R as a subset of R', where a bundle a E Rª is at least as good as a…

A: To write: A preference ⪰ on ℝ+2 as a subset of ℝ+4, where a bundle x∈ℝ+2 is at least as good as a…

Q: Let A C RP and rE RP. Prove that z is a cluster point of A iff there is a sequence of distinct…

A:

Q: Consider the following sets A and B. Prove that A is a proper subset of B (Ac B) A = {x € Z:3hezI =…

A:

Q: Let A = {x E Z* : x is a mutiple of 2 and x < 15}. Rewrite A in roster notatlon.

A: Given that : A = {x∈Z+ : x is a multiple of 2 and x &lt;15 }   Roster form : A representation of a…

Q: If K is a compact subset of RP and z is a point of RP, then the set K, = {r+y :y € K} is also…

A: Given that K is a compact subset of ℝp and x is a point of ℝp. The objective is to show that…