let E be a set that baunded. Prove that Inf E = Sup -E) where R, XE E y

Q: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

A:

Q: Question:. Show that R' is a Banach Space.

A: Banach space: A complete normed linear space is called Banach space. Given:

Q: d) Suppose X = {1,2, 3} and Y = {5,6}. The number of elements of the set P(X× Y) =

A: Elements in X = 3. Elements in Y = 2 Elements in X × Y = 3 × 2 = 6

Q: Let T cofinite topology on R and Y={5,6,7} find Ty?

A: We have to construct subspace topology with respect to the cofinite topology

Q: For sets A and B, prove that AU (B - A) = B

A:

Q: Exercise 1. Prove that for sets A, B,C we have C \(AUB) = (C\ A) N (C \ B).

A: We have to show that for sets A, B, C   C ∖(A∪B)=(C∖A)∩(C∖B).

Q: Prove both sides equal If T ≤p and π ≤v, then TUT ≤p Uv. τ υπ

A:

Q: a) ] Let W = span{(1, 1,0), (1,0, 1), (2, 1, 1), (0, 1, –1)}. i. Is (9,0,9) įn W? Justify your…

A: Given that, (i) To check if (9, 0, 9) is in W. If (9, 0, 9) can be written as linear combination of…

Q: When is S \ T = T \ S for sets S, T?

A: Given that two sets S and T. The definition of S\T=a∈S,a∉T and T\S=b∈T,b∉S. Here, neither of these…

Q: Prove that the boundary points of an open interval (a, b) CR are the points a and b.

A: A point a∈ℝ is said to be boundary point of a set A⊂ℝ, if Ba,r∩A≠∅ and Ba,r∩Ac≠∅ for every…

Q: Let X = {a.b.c.d.e) Find the topology r on X generated by f-{{a.b.c).(b.c.d).fc.d}}

A: X={a,b,c,d,e}

Q: # 2. Prove that for all sets A and B, (A - B) N B = Ø. Proof (A Proof By Contradiction):

A: The given problem is to prove the following statement using the method of contradiction. To prove :…

Q: we defined the symmetric difference of sets A and B to be A AB = (A U B) – (A n B) = (A – B) U (B –…

A: The proof of associative law for symmetric difference is given as follows